Inspired by SNESDrunk's "Unconventional Ways to Play Classic Super Nintendo Games" videos. (You should definitely check SNESDrunk out).
Finding alternate ways of playing a game, whether by doing a self-imposed challenge, or just playing a game differently than it was intended, are a good way of spicing up games you've played hundreds of times.
I'll start with the 30 Melon Challenge in Yoshi's Story, where you try to get all of the 30 Melons in each level, and NO OTHER FRUIT. It's basically like Hard Mode for the game. Particular Mushroom has a video that goes into more detail.
It's a minor thing, but whenever I get into a pokemon game, I deposit my starter ASAP. I don't want to rely on the pokemon which is spoon fed to me from start, it feels a lot better to play the game with whatever I'm able to catch along the path.
My first time playing Pokémon Blue I did the opposite because I didn't realize you could change the order of the pokemon so my Bulbasaur was in my #1 spot and even if I knew the other trainer was going to use a fire pokemon i'd end up starting the battle and wasting a turn switching pokemon. Needless to say he was super overpowered by the end of the game from all the forced XP
I did the same way back when playing pokémon red. My Blastoise was up in level 60-70 when I approached the last gym. Being a small kid, I didn't understand tactics of effective types - only that my pokemon was an awesome heavy turtle with cannons on its back. I never really tried to use other pokemon or switch out during battle.
I tried to play Ultra Sun using only my starter Rowlet, and had to put it down when the game forced me into a double battle. I would have been okay with hatching a second Rowlet from an egg but it didn't work.
I love me an underrated Bug superstar. In Black I used a Leavanny as mine and it cleaned house. When I played Omega Ruby my Masquerain carried my team and took me totally by surprise.