My favorite evolution method is traditional leveling up. I don't really like evolutions that rely on hard-to-get or one-of-a-kind items, as I don't like to use said items unless I've bred a perfect Pokémon to use them on, and that isn't very practical during my initial play-through. Leveling up is straightforward and predictable, and can be done anytime, anywhere, with just a single game and system.
As I mentioned above, I don't really like rare-item-based evolutions, and in particular, I don't like trade-while-holding-an-item evolutions. These involve not only using up a valuable item, but also require you to find another player to trade with. While trading is easier than it once was (I recall having to wait until my only Pokémon-playing friend came over and having to carefully hold the link cable in place during trades), it's still tedious, and for players who don't have multiple systems, depends on trusting strangers on the internet. I know Pokémon is supposed to be social, and that's part of the reason for trade evolutions, but realistically, I don't know that many people who play Pokémon, and even when it was all the rage, very few people I knew actually wanted to spend lots of time trading anyway.
If I were to create a new evolution...hmm, I'm not sure. There could of course be expansions on GameFreak's existing ideas (e.g., new stones, evolutions at different times of day (perhaps more precise than the existing ones, though that would be frustrating), evolution based on relative stats at a certain level a la Tyrogue, new trade evolution items, etc.). There could be evolution when some particular stat exceeds a certain value for the first time, although that wouldn't be that different from level-up evolution (it would just happen over a range of levels rather than at a single one). There could also be evolution that depends on catching a certain Pokémon while having a different Pokémon in the party (i.e., while Pokémon A is in the party, Pokémon B is captured--> Pokémon A evolves). There could be task-based evolution (e.g., Pokémon A evolves when it successfully solos the Elite Four, or completes some other task). There could be Daycare-based evolution (put Pokemon A in the Daycare with Pokemon B--> Pokemon A evolves when withdrawn. I'm sure that there are lots of other ways GameFreak could have Pokemon evolve, but I'm not sure that they're any better (or even as good as) most of the existing methods we have for evolving Pokemon.