I like Azelf simply because the name sounds cool. And it's AWESOME at competitive. In fact, the other two are not bad either. In-game they are sort of like misc. legendaries who make up the Red Chain for big bad brute called Cyrus. Yes, I'm Cyrus serious. Mesprit in particular is a pain to catch and get a good one.
The dudes themselves seem nice and likeable, and the lore behind them makes a good background. I think they are so-so, however, so I don't really think much of them.
Their designs, as a matter of fact, are awesome. I like the fairy style and I do hope they get retyped (jfthoi) as Psychic/Fairy (maybe already are? idrk cuz I didn't hear the recent news) and so become fairy.
They always do seem as chibi versions of some hawt fairies to me, too. Don't mind that, I'm just that kind of an introvert.
Of all the trios, the Lake Trio is my least favorite. They had a big role in the games, but not so much in the anime. I do like their position in Sinnoh's mythology, however. I'd like to know more on how they granted everyone knowledge, emotion, and willpower. Uxie is my favorite of the three. I'm not sure why, but that's the first Pokémon that pops in my head when I hear or see Lake Trio (which is strange because I like Mesprit's name and Azelf's design more than Uxie's).
While they aren't my favorite trio, I do like them quite a bit for their designs and what they represent. Their biggest drawback for me is the boring typing and such similar designs, but overall they aren't bad Pokemon. My favorite of the trio is by far Azelf because of it's sheer strength. This Pokemon is an excellent battler because of high speed, massive offensive stats, and great movepool. Uxie is second for me because I love what it represents and it has the best design of the trio. It can also serve as a really nice support Pokemon in competitive play. Mesprit however, is just ok. There is nothing too special about it at all.
I really like the concept behind these trio—how they bring balance to the Pokémon world by keeping the power of knowledge, emotion, and willpower in check. Their designs are pretty simplistic, which I love. They do look quite similar to each other, but I'm all right with that.
My favorite of the lake guardians would have to be Azelf. Its primary color is blue, and I'm a big fan of that. :) When it comes to battling, this Pokémon can perform quite awesomely. It's got a great Speed stat, which gives it a chance to outrun many Pokémon in its tier and land the first hit. With its above-average Special Attack, it could take down your opponent's entire team single-handedly after just one Nasty Plot. Azelf can learn Stealth Rock, which would make it a great lead. After you have Stealth Rock set up, your opponent's Pokémon would wreck into them as soon as they're sent out on the field, making it easy for you to finish them off. This Pokémon also has access to Reflect and Light Screen. Give it a Light Clay, and it would make an excellent dual-screener for just about any of your team!
I've always considered Emotion one of my strengths (and weaknesses) so Mesprit is my intuitive favorite based on their ideas. I think Azelf looks a bit better though, so it's difficult to choose between them. I don't much like Uxie's closed eyes.