I have been an AD&D 2e DM for many, many years. However, I recently decided to update to D&D 5e when I concluded that the rules in 5e — especially overall — are more to my liking. Unsurprisingly, 5e is a more polished game than AD&D 2e; this makes sense given that D&D in general has been workshopped far more by now than it had been when AD&D 2e was first created.
With that being said, in AD&D 2e, there was some really fun, crazy stuff that went down. For example, the point system for selecting non-weapon proficiencies was SO fun, beloved by all of my players and I. Because of it, players could have some of the most hilarious combinations of skills; I had a character once with the Gaming (like poker, blackjack, etc; fell under Charisma) proficiency, Astrology (reading star signs and predicting the future) proficiency, and Engineering (building cool stuff, weapons or otherwise) proficiency, and managed to combine all three in a couple situations.
In 5e, there are skill proficiencies with associated modifiers and corresponding ability scores, but there seems to be very little player choice in selecting those proficiencies. Sure, the backgrounds offer a little wiggle room as they can be customised, but nothing like the hilarious lunacy that AD&D 2e provided. Additionally, there seems not to be a way for characters to acquire additional skill proficiencies as they level up; in AD&D 2e, the non-weapon proficiency point allocation system awarded points (to be spent on new proficiencies) to characters when they levelled, and choosing new proficiencies was part of the fun of advancing as a character.
Not only that, but as a more general D&D complaint (as there's never been a good system in place for this, in my experience as a player and a DM), choosing to train in (up/boost/improve modifiers for) specific skills or abilities seems to have no place in 5e, nor in D&D in general. We had house rules that allowed players to "train" in specific skills or abilities (the 6 basic ones, plus a new d20 based ability score called "Chakra" that was analogous to mana) during pre-determined training periods, using Naruto as a model for this. In fact, we had several campaigns wherein we overhauled the spell-casting system to be Chakra based, essentially meaning that one factor determining the player's ability to cast spells at any given time was based on a fixed, decrementing mana value, and gaining new spells was based less on level and more on specifically training to learn specific spells, or "Jutsu" in those games.
TL;DR: Is there a way to give players more choice in what skills they do or do not have, both in game (as they level up) and during character creation?