I was playing, some decades ago, a D&D 1st ed. homebrew campaign using my human mage player.One of my teammates ( a female Monk ) asked me if my character had ever been in love, and was so curious to know all the details. I used a wooden stick to write on the ground the name of that specific girl, in elfish idiom. The monk didn't have proficiency in writing/reading that idiom and asked the DM if she could guess what is that name.

The DM said: ok, roll an Investigation check. How funny it was when the monk rolled a Nat 20, and the DM allowed the Monk to know the name I wrote on the ground. Hours later, when the session ended, the DM said to the player who was using the Monk: If you continue rolling Nat20's on this Investigation roll, for at least 9 days more, I'll allow you earning  this new idiom you don't even have proficiency with.

So, since this new 5th edition don't have a similar rule for those strangely lucky rolls, can a DM allow any character do the same ??

Can a dm allow it? Yes a dm can allow anything that they think enhances the game.

If you are looking for related rules, I think there are downtime language learning rules.

I'd second BKW.

Statistically you need to roll 20 times a day for 10 days. That sounds very much like grinding and, for me at least, really not fun. Why woukd you be rolling an Investigation check on this one language 20 times a day for 10 days straight? Surely, there's a better way to do it. If the DM wants to do it this way, don't make it known. He should have just declared it after the fact: "What? You rolled Nat20s every day for ten days? No way! Tell you what, due to your amazing skill, you have now learned..." By declaring it first, there made it a goal, and hence a grind to get it, rather than a reward.

Also, I'm very unsure of what you mean exactly by "idiom". Your usage of the term isn't in line with its meaning.

I think to know the dialect, one has firstly to know the idiom-root which it comes from, right ?? So, taking this in mind, spending 250 days it's a bit more of a 2 third parts from a whole year, only to know the idiom. Later, after learning the idiom-root, it could be easier learning the dialect, spending the last third part of a year.

Quote from Linklite >>

Statistically you need to roll 20 times a day for 10 days. That sounds very much like grinding and, for me at least, really not fun. Why woukd you be rolling an Investigation check on this one language 20 times a day for 10 days straight? Surely, there's a better way to do it. If the DM wants to do it this way, don't make it known. He should have just declared it after the fact: "What? You rolled Nat20s every day for ten days? No way! Tell you what, due to your amazing skill, you have now learned..." By declaring it first, there made it a goal, and hence a grind to get it, rather than a reward.

 The way i read the original post I would expect the option of 1 roll a day and if you break the chain of 20's then you don't learn the language. (feels very you can try but very unlikely to succeed)

But for the original post it looks like the consensus is:

1. there are official rules for learning a language that require significant downtime
2. there is nothing preventing a dm in 5e from implementing the scenario in the original post if they wanted to implement a house rule.

I prefer hard rules around this kind of thing to avoid a - to use an idiom myself - squeaky wheel gets the grease situation.

Some people are always going to wait patiently for "official" downtime to try to gain minor mechanical bonuses like languages or new skills. Other people will just ask for it or try to bargain for it. I have seen resentment grow at a table where the DM (reluctantly) gave way to this kind of pushy player resulting in that character consistently gaining more benefits. It wasn't so much favoritism as it was the fear of saying no and seeming like a jerk. It made the rest of us uncomfortable, and we didn't push for the same thing because it just felt... whiny.

This is the kind of thing rules are made for. "Sorry, it takes X time to study with Y resources and right now you're in the middle of a tomb swarming with undead. Maybe you can make some progress on this when the adventure is over and everyone gets a chance to spend some downtime."

Rules don't have to make real-world sense.  For instance, a rule might say that a tiny creature can sustain the same extent of falling damage as a gargantuan creature which, in real-world terms, is nuts.  If a DM, for whatever reason, agrees on a rule, then can be a hard rule.  If the rules say that you can't fall while you believe there's something under you then that's the rule.  It's just a different kind of world being built.  The important thing is that the players know the rules, whatever they are, and can all have relevant access to them.

in general though, rules that can allow players to gain buffs beyond those of other players have the potential to create party imbalance and these rules could be avoided.  Having said that I like the idea of bringing the intelligence stat more into play as it is frequently dumped.

A DM can allow whatever they want in their game. AD&D 1e didn't have an investigation skill (it didn't have ability checks and skill proficiencies either) so the DM was already effectively using his own homebrewed rule for that so there's no reason they couldn't do in it 5e. 

It’s already covered in Tasha’s I believe. 
10 week base to learn a language +/- your Int modifier. 
so an 18 Int learns it in 6 weeks. 

6 weeks ??? woow... a nice sidequest to be done while inbetween campaigns, or while in a long travel.