I have a few favorites, all of which I love and none of which I can quite put as absolute #1's. Sadly, most of them are my favorites due to the Pokemon they use or their design, since I'm kind of emotion blind when it comes to text.
Wallace: I adore Wallace. Largely for more... personal reasons >>;... but I just love his look and his style. He looks kind of like the sort of character I'd design in a video game, or how I'd want to look as a trainer, and he specializes in my favorite type. He also seems to have a really good personality.
Glacia: There's not a lot to say about her since she's pretty much just an Elite Four member... but she's just a wonderful design... in the remakes, at least. She specializes in my favorite element (NOT favorite type), she has a beautiful look to her, and her Elite Four path and room is amazing. She's also reasonably tough for using one of the biggest pushover types. She's also one with some... personal preferences.
Lorelei: Similar reasons to Glacia, though not quite to the same level. As she appears in Gen I and Gen III games... her team options are a lot more limited than later Ice trainers, so it isn't QUITE as interesting, but she gets extra points for being the ORIGINAL Elite Four member in the series. She's the introduction to all of it, and she shows Ice -can- be formidable.
Drake: He's an old sailor with a cool moustache that specializes in Dragon-types. There's honestly nothing else to say about why I think he's awesome. lol
Side note edit... As for Joey himself? I've always had kind of a... weird connection to him. Being named Joey myself... it's always felt kind of awkward, especially considering he's technically fairly obnoxious. But I love him for some weird reason, and I love making Rattata jokes because of it.