Ah, I'll join the bandwagon! I've never used an eeveelution in battle though, so I'm gonna base this only on their design and dex entries. And how I think it'd be like to have them as pets ^_^
So, going by dex number:
Vaporeon - very pretty and elegant, I think it would make a great swimming buddy, though you'd need to have a pool to keep one, I can't imagine it being able to stay out of the water for too long.
Jolteon - free electricity, yay, as long as it doesn't shock everything that dares to touch it. Not that it looks very huggable, hehe. Actually, it reminds me of a guard dog.
Flareon - very cute, so cuddly and fluffy! I would rate it the best hugging buddy! I've heard that people look down on Flareon because of its battling prowes, and it's a shame, it looks like such a great Pokemon.
Espeon - I think it looks amazing, kind of mysterious actually, with just a bit of creepy. I read that its fur is supposed to feel like velvet to touch, so maybe it's not that scary to be around it. Also, it's worth noting that, since this is a cat Pokemon fan club, out of all the eeveelutions, Espeon definitely looks the most feline to me.
Umbreon - it has a really cool design, I'll give you that. It looks mysterious, ready to jump you from the darkness of night, great in battle, it's just... it was never my favorite, I don't know why.
Leafeon - this is my favorite design, I think. It's very cute, and looks, I don't know, loyal and trustworthy? It just gives me that sort of impression. And it is said to purify the air, so it would be nice to have it around, fresh air, aaaah.
Glaceon - very elegant, almost haughty. I think that's the reason it throws me off a little, because otherwise it'd be among my favorites. Looks very fitting as an Ice type, that's for sure. Nice to have around during those hot summer days, hehe.
Sylveon - oh my gosh, its design just has so much going on, right? I'm really curious to see what kind of personality they're gonna give it, right now it looks like a happy Pokemon that loves to eat candy ^_^
Also, this took way too long to write, I even ate dinner half way, hehe.