I'd have thought that saving throw proficiencies might be dependent on a range of issues in a character's background. For instance, I can quite imagine that a druid with an outlander background or a wizard as a hermit in some isolated location or who has previously dabbled in alchemy might both start with a constitution saving throw proficiency.

It also seems reasonable to me that a character that starts with a level in one class but then moves to several levels in another class might have their best saving throws changed to those of the new class.

Alternatively, it also seems reasonable to me that a character's saving throw proficiencies might correspond with their highest ability scores.

Whichever way perhaps there could be some flexibility in saving throw proficiency distribution.

At present players can be tempted to drop potentially preferred character backstories for the sake of starting with a class that gives the saving throw proficiencies they prefer.

While I don't disagree that the class saving throw proficiencies are somewhat arbitrary, I do disagree that changing them based on background, new class, or highest ability scores should be allowed. Constitution proficiency should be rare. You should only get one of the big 3 as part of your starting package. You have options to fix this problem in game now, if you think they are too expensive, then make those options more accessible rather than trying to "fix" the system by such a technique.

Quote from GergKyae >>

Alternatively, it also seems reasonable to me that a character's saving throw proficiencies might correspond with their highest ability scores.

Having a higher ability score is already giving you a better saving throw; there's no reason to drag proficiencies along for the ride

Quote from GergKyae >>

At present players can be tempted to drop potentially preferred character backstories for the sake of starting with a class that gives the saving throw proficiencies they prefer.

 Honestly, this feels like an easy conversation with your Dungeon Master.

"Do you mind if I swap my Saving Throw proficiency for <reason>?"

It's less about changing the rules to the game as a whole and more about partnering as a player and DM on what makes an interesting character for a specific campaign. I do this all the time with my players. I grant other DM's might be less pliable and protective of RAW, but it's worth a conversation to find out.

tl;dr: Seems easy enough, but I wouldn't want it to be in the next major sourcebook of rule additions and changes.

There are feats that allow you to take proficiency in other saving throws so they could get hold of them that way. It also gives you +1 to that ability score. Could be a good thing for them to have.

I'd be willing to allow players some flexibility in their saving throws, but I'd stick to the current "rules" on what you can do with it.  So one from dex/con/wis and one from str/int/cha.

Feats for most characters only come in to play before 4th level in a class. 

I think it would be fine to ask a DM whether your archer based fighter could have, say, DEX CHA saving throw proficiency similar to a Bard if those were the abilities they'd focussed on in their development.

I'd say that players might reasonable ask for one out of DEX, WIS and CON and one out of STR, INT and CHR in any combination.

I'd have said that if a player really wants a character to have a particular saving throw then chose a class that has them or pick the resilient feat (if you're playing feats) when you get the choice.

While this sounds draconian and inflexible I can see CON becoming a de-facto choice for any spellcaster for concentration checks if you give players a completely free choice. This might just be paranoia (or projection) on my part though!

This means that characters would either have a choice of starting with artificer or fighter for wizards, barbarian for druids or sorcerer for warlock.  Either that or choose variant human or, if allowed, custom origin rather than one of the rich range of d&d races.  
In many cases, it would make sense for a wizard to start off as an artificer but I don't see why, say, a dwarven wizard couldn't start directly with INT and CON saving throw proficiencies without needing to wait 'til level four.

As I said, I wouldn't allow it for the reason I outlined. If it offends your sense of what reality is in your game-world then change it.

Con for spell casters is a significantly beter save proficiency than anything else.  I might be wrong but I believe that sorcerers got con proficiency as part of the balancing with other classes.  Therefore I think a paladin or a cleric swapping wisdom for Con for free is a free boost to their power and I would not allow it.  If a Dex based paladin thought it suited their character better to have proficiency in Cha and Dex however I might allow it.

Where it gets odd is with multiclassing. Anyone planning to be a sorlock will take sorcerer first  for the con saves, a warlock that somehow manages to get out of their pact might think it is most appropriat for them to have developed an inate magical ability as a result of what the patron had given them and start taking levels of sorcerer. You might then end up with the sorlocks at level 12 both with 5 levels of warlock and 7 of sorcerer with little difference except the "planned"  sorlock has con saves and the warlock that got out of his pact has wisdom

Character origins begin with their race and then, presumably, their backgrounds.  None the less I'd agree that it could be hard particularly to justify swapping any cleric's saving throw proficiency away from wisdom which seems typically to be central to their training.  Not sure if the same would necessarily apply to potentially rugged, life in the wilds druids that don't face futures wrapped in metal.
In the case of a Lizardfolk who had something like a marine background prior to their encounter with a fathomless entity, I think there might be a case for saving throw proficiency in CON and CHA. 
One of my first 5e characters was Dan Galf the dwarven wizard.  I never could figure out why he didn't have con saves!  In this case Dan had 12 wisdom from point buy yet, in comparison to his constitution, this was still far from his strength.

Quote from GergKyae >>

I don't see why, say, a dwarven wizard couldn't start directly with INT and CON saving throw proficiencies without needing to wait 'til level four.

Why would they have proficiency in a CON save, as opposed to just having a higher CON?

Dwarves are all about constitution, as might be a Lizardfolk would be warlock who had that marine background.  
I think situations like this might at least warrant a question.

Quote from GergKyae >>

but I don't see why, say, a dwarven wizard couldn't start directly with INT and CON saving throw proficiencies without needing to wait 'til level four.

 
For game balance reasons. Proficiency in Con saves is super powerful, especially for spellcasters, and the Wizard is the single most powerful spellcaster in the game. Sorcerers get proficiency in Avon saves because they need the extra boost to be competitive. If people could pick and choose their Saving Throw proficiencies then everyone playing a Spellcaster would pick Con.

To be honest, having proficiency in a Save associated with a lower Ability score is very useful because it compensates for the lower raw number and makes the character more all-around survivable. For examples:

• Most folks who make martial characters go with a Dex build because it’s more efficient. Str allows for bigger weapons and heavier armor, but heavy armor is a pain in the batooty, and there is only 1 Skill associated with Str. So, many folks dump Str and focus on Dex (for skills, AC, and Attacking), Wis (fo’ da skillz & saves), and Con (for their HP and saves). If you dump Str then you got nothin’ in the saving throw, but knowing that proficiency will kick in, that isn’t a big deal for those Dex Fighters.
• Most Wizard PCs are already strong in Int based stuff because derp, so they don’t always worry about stuff like Perception or other Wis skills. And they all want Con for Concentration and Dex for AC and when they actually need to use their dagger. To that end, many Wiz players don’t prioritize Wis as much, especially knowing that many of the others in the party will. So having PB in Wis saves and having higher Dex and Con anyway also makes a Wiz more all-around survivable too. 

Quote from GergKyae >>

Dwarves are all about constitution

Which is why they usually have a higher CON score

You haven't provided a reason why they would be proficient in CON saves on top of just having that typically higher score, which is already giving them a better save bonus

Quote from AntonSirius >>

Quote from GergKyae >>

Dwarves are all about constitution

 

Which is why they usually have a higher CON score

You haven't provided a reason why they would be proficient in CON saves on top of just having that typically higher score, which is already giving them a better save bonus

 It just seems strange to me that a dwarf, whose essence is centred on constitution and resistance, has to wait until class level 4 to gain proficiency in con saves.  

It seems strange to me, if you have a build plan for a wizard that utilises some levels in a class like fighter, that you have to start with fighter in your wizard build for the sake of the saving proficiency.

It might seem strange, because you are looking at it from a strange angle. What you come from only has a little to do with what you are (hence the stat bonus); most of what you are is what you've done with yourself. Your class provides you with the proficiencies that you would gain as you become a proficient member of that class.

Quote from WolfOfTheBees >>

It might seem strange, because you are looking at it from a strange angle. What you come from only has a little to do with what you are (hence the stat bonus); most of what you are is what you've done with yourself. Your class provides you with the proficiencies that you would gain as you become a proficient member of that class.

  I find it strange that a fighter 1/wizard 19 with a sage background does directly get proficiency in constitution saving throws but, for instance, a wizard 1/fighter 19 with a hermit background, if the levels were gained in that order, doesn't.
In the PHb we read about:
"Backgrounds
Every story has a beginning. Your character’s background reveals where you came from, how you became an adventurer, and your place in the world."
This forms the foundational story of, "what you've done with yourself".  
Fine if you want to describe getting to level 1 in a first class as some kind of mystical, magical event (d&d level ups are full of this stuff at level ups), but otherwise no, it does not make sense. 

Quote from GergKyae >>

It just seems strange to me that a dwarf, whose essence is centred on constitution and resistance, has to wait until class level 4 to gain proficiency in con saves.  

Would you rather be a level 1 dwarf with an 18 CON, or a level 1 dwarf with a 14 CON and proficiency in CON saves?

They end up with the exact same bonus on their CON saves, but the former seems a lot more like a character "whose essence is centred on constitution and resistance" than the latter to me.