I have a rule where my players can spend downtime and gold to increase their ability scores the same way they would get a new skill proficiency or something similar. The cost and time increase depending on what number the skill is (going from 8 to 9 STR is going to take a lot less time and effort than going from 14 to 15 STR). I don't see a difference between this and the rules in the book about gaining XP in downtime. 

That said, I do not remind my players of this rule. A Wizard in my party also wants to increase his strength and makes sure to tell me about his exercises he is doing before they rest and whatnot, but when the downtime comes he always has something else he wants to do so he has yet to actually take advantage of this system (though my other players have). 

If I were to do something like this, it would be to alter your stats, not to raise them.  Meaning you could focus on raising one stat to get it up, but in the time you were doing that, another stat would lower from neglect.

I agree with the others that the ASI is what should represent this sort of activity - if they put theirs into intelligence, then this means that they spent their time reading books, not working out.

Now, one option would be to make this into a plot point. How badly do they want the strength? Would they offer their services to a greater power to get it? Perhaps someone or something has been watching them, and knows what they want? Here's my pitch:

Open with telling them that they can spend their downtime trying to locate a pair of Gauntlets of Ogre Strength. They can do research on it, finding last known locations, etc. Whenever they do so, make them roll for insight but keep telling them the trail runs cold, and they can't find them.

Then have a stranger visit them in the night, and tell them that they heard that they were seeking these gauntlets, and offer them to them - perhaps stylized as supple leather gloves, more suited to a wizard. They offer these to them freely, saying "there is no price - not yet." Do all this in private so the other players aren't shouting at them to say that it's clearly a trap and they are selling their soul to the devil.

Then, after they have sold their soul to a devil in exchange for magic gloves, you can apply a curse which isn't detrimental to the Wizard but is to the parties efforts - namely that the Devil can, at will, see & hear using the wizards senses and so learn everything they discuss, where they are, etc., and will lend their assistance to the BBEG in exchange for souls of their prisoners (or whatever works for the BBEG in your campaign). The BBEG starts to pre-empt their efforts, making things more difficult for several sessions.

Then, in the background, the Devil oversteps and offers the services of the wizard to the BBEG in exchange for more of what they want. The Devil appears to the wizard whilst they are alone and tells them that they have a task for them, which they explain directly. The wizard will almost certainly refuse - the Devil offers nothing, simply saying "I own you!" - and this will lead to the Devil seeking to kill the wizard and take back the gloves. The ensuing fight might be close enough to involve the whole party, or it might take place later, but the result will either be a dead wizard or a dead devil - the gloves either earnt through the story, or paid for in full.

To my mind rather then allowing the wizard to work on Strength I would allow him to work on specific skills that are strength related, to equate this to a real life example a friend I know is a body builder, over years of effort  training focus and dedication, going to the gym twice a day 6 days a week he has genuinely increased his "strength attribute" from his base level, but, that has come at the sacrifice of other aspects of his life due to the time he has to dedicate. 

Another friend goes climbing 3 times a week, he is fitter and more athletic than he was, but I would not say overall his strength has significantly improved, he has improved his skill at climbing and he has a more varied life with time to dedicate to other things. 

A wizard has things they need to do daily, preparing spells, reading and putting spells in the book, studying, researching, preparing spell components. If your player insists he wants to buff his strength outside of his normal ability then tell him it is at the cost of something else, the sacrifice required will cost something. 

The whole point of the game is balance, you sacrifice one area to be strong in another, or you become average across them all. The party then balances this out with the wizard being good at intellectual abilities, the barbarian being good at strength etc. I would take probably a slightly harder approach to this and talk to my player and explain simply that no, they can spend time becoming proficient in a strength skill, but they can not increase the attribute  if they are so dedicated to mapping out the perfect wizard, they need to expect their strength to be low, and then use it to lean into for roleplay purposes. 

This may sound harsh, but I would not allow it even if I am otherwise very generous in many respects.
The request undoubtedly comes from the player (and not the character) and is based on the mechanics of the game that the player has in mind.
Now this is not meant to be a powergamer vs roleplayer argument, because these characteristics are not mutually exclusive. But good roleplaying starts when you understand that the character you are playing is a person.
A mage, for example, is a person who dedicates their life to magic. That means studying and learning, and there's little time for bench pressing. Of course, you can also create a less stereotypical character and give a wizard the feat Tavernbrawler and a Strength value of 14-16. That way you have a person who may actually spend a lot of time on physical activities, but at the expense of other attributes. This fact not only balances the game but also gives credibility to the world and the characters.

If the player cared about his Wizard STR score, he would have spent points on it in point buy, or assigned a higher score to it through rolled or standard array.

Why is it so important that he wants to get it for "free" but can't be bothered to spend resources on it?

Kinda just sounds like your player wants to be better at strength saving throws and checks for free.

Quote from Lunali >>

Quote from Tim >>

I would put together a quest for the party to find a magic item that increases strength. Maybe a https://www.dndbeyond.com/magic-items/ioun-stone-of-strength.

 Since they both require attunement anyway, I'd go with the lower rarity gauntlets of ogre power.

I would, but the gauntlets give a flat 19 Strength. The Ioun Stone just increases it by 2 to a max of 20. Even though it’s more rare it’s less of a power bump to the wizard. 

This is a hard no for me. It is what ASIs are for. Don't want to spend your ASIs on Strength but on something else? Then I guess Strength wasn't all that important to you after all.

You simply cannot allow this for him, without allowing the Barbarian to spend down time in a library to improve his INT, the Cleric to spend time on the balance beam to improve her DEX, the Fighter to spend time talking in front of a mirror (a la Sims 2/3/4) to improve his Charisma. And so on.

Either every PC gets to up their stats for free, in addition to ASIs, or nobody gets to do it. You should never allow this sort of thing for just one player. It is not fair.

Rather than a daisy chain of "you get a quest and you get a quest and you get a quest," the party learns of the tomb of Ali Versatile Swissknife, resting with them in that tomb are all the items the party seeks for this rejiggering of everyone.

Ali Versatile Swissknife was a Cleric2/Druid2/Wizard2/Fighter3/Rogue3/Palladin3/Ranger3/Warlock1/Barbarian1 who rolled a 13 in everything, may they rest in all around competence until the party finds utility in commandeering their remains.  Maybe in addition to the magic items, there can be relics  from his remains that would be iterations of "body part of decency" each relic granting a 13 in a particular ability.

Quote from nat1allday >>

 

So, I have decided to let each of the players decide on one magic item of uncommon rarity or lower, and the entire party will quest for it together. I will just create a couple mini quests and randomly generate an order for them. At the end of each of them will lie the magic item they wanted.  

 This can be good and bad. It can give your campaign some direction but I'm not sure how much I would enjoy it as a player. But that's just me. Some folks will really get behind the quest for the magic thingie knowing what the magic thingie is ahead of time.

Instead of opening the conversation with "What do you want?" I see if they will ask, "Is there this thing?" The difference being player initiative.

Quote from Wysperra >>

Quote from nat1allday >>

 

So, I have decided to let each of the players decide on one magic item of uncommon rarity or lower, and the entire party will quest for it together. I will just create a couple mini quests and randomly generate an order for them. At the end of each of them will lie the magic item they wanted.  

 This can be good and bad. It can give your campaign some direction but I'm not sure how much I would enjoy it as a player. But that's just me. Some folks will really get behind the quest for the magic thingie knowing what the magic thingie is ahead of time.

Instead of opening the conversation with "What do you want?" I see if they will ask, "Is there this thing?" The difference being player initiative.

 I ask my players what they would like to enhance, the rogue might say stealth, the fighter might ask for a better weapon. I will then use that to stock my magic shops, not with everything they asked for, but the sorcerer might want bracers of defence and instead find a cloak of shielding. It gives the sense that the world is not amazon, get what you want when you want it. 

Quote from nat1allday >>

Quote from Bodanger >>

Essentially what the player is asking for is a free ability score increase.

Granted its not like it's for the casting stat or anything but still.  Are you prepared to let the entire group do this kind of thing?

I'd consider it  but there would definitely need to be a cost and first I'd need to know why.  Is he doing it to meet the str min for some kind of power gamey multiclass like trying to meet the min str for armor or something, or is it just for flavor/ str checks?

There needs to be a cost, something like allowing him to swap out dex, or next level up he learns no new spells due to the time being used to work out instead of studying.  Another thing you might consider is something like gauntlets of ogre strength which would take up an attunement slot.

 He is literally just uncomfortable with the concept of a puny anorexic wizard. The no new spells idea is actually pretty smart. Or perhaps actual gold or something. Everything has a cost in life, so the same should be true in D&D. The gauntlets could work as well, especially since other buff players won't want to fight for it because they already have a 20 in strength

 Since Forgotten Realms D&D doesn't have much to spend money on,  it would be a bad idea to let them just buy higher stats because they would be able to easily afford it.

Also, let's remember low strength doesn't have to mean puny or emaciated (probably a better word than anorexic).  People can lack "strength" for a lot of reasons besides low mass.  Maybe there's a back problem meaning they can't get the whole dorsal chain of muscled from upper back to calves activated for the big lifts (which would indicate compromised ability to deliver power with other muscle groupings and chains).  Maybe they're average build but live a relatively sedentary or lounging life so again lacks the slow and fast twitch fibers as well as practice in the kinetics behind delivering strength.  Maybe they're on the overweight side and lack the aerobic capacity to support anaerobic effort.  There's lots of shapes and sizes a "not strong" PC can come in besides the "puny" stereotype.  So if the players concern is really just annoyed with characterization of their stats, draw them in a different light.

He is literally just uncomfortable with the concept of a puny anorexic wizard.

Then why did he make one up? Did you force the low strength on him?  If so, then maybe I can see it, although I would suggest he should have said that from day 1, not now.

Also, low STR does not mean puny and anorexic. If he's RPed his character that way, again, why? Did you force it on him? Are other players RPing that he's puny and anorexic even though he'd rather not RP that?

It sounds like some of these are HIS choices and now HE is not happy with HIS OWN choices -- the choice to put his lowest stat in STR, the choice to RP that he is a puny anorexic. If these are HIS choices, then as a player HE needs to learn to live with HIS OWN CHOICES. And maybe learn the lesson that choices matter - not just what stats you have but how you choose to RP those stats.

I'm not saying he needs to suffer. But I would have a conversation with him at this point about how as DM, yes, I am willing to work with him to help him make different/better choices going forward, and maybe help him deal with the choices he has made up to this point, but that, in the future, he needs to think carefully about the choices he is making and recognizing that part of an RPG is making choices, and then living with those choices (or dying from them, in some characters' cases). After all, when he lights up a fireball that kills innocent bystanders, THAT is a choice too. He's got to live with that as part of the RP as well.

These games are all about choices and consequences. That's what makes something an RPG. If you want to play a game without choices and consequences, there are board games like this (e.g., Candyland, where everything is literally random and there are no choices to make at all). But RPGs are not like that.

Quote from nat1allday >>

So everyone has repeatedly said that it's a bad idea, and that I should either make him do one of the following:

1. Quest for an item like the gauntlets of ogre power
2. let him do it but at the expense of another stat
3. Provide everyone with the opportunity to do this.

So, I have decided to let each of the players decide on one magic item of uncommon rarity or lower, and the entire party will quest for it together. I will just create a couple mini quests and randomly generate an order for them. At the end of each of them will lie the magic item they wanted. Lastly, I will also provide all of them the chance to get an unlimited number of +1 bonuses to any score they please, but for each +1 bonus they get -2 to a score, to a minimum of 4.

 Be careful with this, Gauntlets of Ogre Power plus your houserule means the wizard can further dump his STR to 4 in order to power up other stats and still have an effective 19.

Some people like to play in superpowered games where everyone has extra stats and no one has a weakness unless they want one. You may want to consider pitching that to your players if that's the game they're looking for. As long as everyone is roughly the same power level it's not too hard to ramp up difficulty to meet it.