Okay, here we go. Best is a very relative term, but I assume you mean best for a playthrough, because that's what you do in a main series game, so I will try to give you the best one for a playthrough of that specific game. It does depend on the rest of your team, but I'll do the best I can.
RBY: Use Vaporeon, as surf is a needed HM, and Gen 1 Blizzard is OP. Also a Special stat of 110 combined with 130 HP makes Vaporeon an amazing special tank. Better than Flareon, because Flareon doesn't get any good physical moves for it's 130 Atk and Jolteon has a way too shallow movepool. Vaporeon also has good matchups against every boss except for Misty, and you can't really get an eevee that early game.
GSC: Use Espeon, because 130 Spa is amazing, 110 speed outspeeds most ingame trainers because of EVs, and STAB psychic rips through everything. Also gets bite, which is Special in gen 2. Better then Umbreon, because the only thing Umbreon can do is toxic stall, because he has horrible offensive stats, and toxic is a very late game TM anyway. Better than the other Eeveelutions because it's very, very hard finding Evolution stones in the gen 2 games.
RSE: You can't get Eevee in this game until postgame, so by the time you get one, you're already done with all bosses except for the battle frontier and Steven in Emerald. If you still want to use an Eeveelution against the battle frontier, Vaporeon can run double team + baton pass and learns surf and ice beam too.
FrLg: pretty much the same as RBY, but use Ice beam over blizzard, because blizzard has bad accuracy now. Vaporeon loses it's 110 special defense, but is still tanky.
DPPl: You can get an Eevee in the trophy garden, but it's a daily Pokémon, which make things harder. Physical and Special split happened, so Flareon can finally get a good fire STAB, right? Worng! he only gets fire fang, which is a sad 65 bp. Leafeon is available pretty early, but learns leaf blade, it's best move, at level 71. Use Vaporeon, because it's OP, easy to get, and has a great movepool. Jolteon is also a viable choice, with charge beam now here and it finally gets coverage moves, in the now special shadow ball and signal beam.
HgSs: Unlike GSC, Vaporeon is now better, with other special moves like shadow ball and signal beam, also Psychic TMs are a pain to get, I have personal experience with this. Also, there's no great grass or electric trainers until Kanto, so you have good matchups on bosses. Ice beam can help with Clair.
BW: No Eevee here either. you have to migrate one from a gen 4 game. you could use one in the battle subway I guess... Does Flareon have flare blitz yet? of course not. Use Espeon, same amazing stats, same great movepool.
BW2: Eevee is now available for your playthrough! and early too! Avalable at Castelia city, home of the 3rd gym. Evolution stones are easier to get, and Vaporeon is good as long as you don't try to use it on Elesa. Ice beam/Blizzard will help with the dragon gym leader and the dragon champion. Espeon is weak to two of the elite four, so not as good.
XY: So far it has been pretty one-sided with Espeon and Vaporeon, but now we Flare Blitz Flareon! (Just kidding! with 65 hp and 65 speed he gets outspeed and OHKOed) now we have Sylveon! Eevees are easy to obtain, and Sylveon learns moonblast at level 37, which is freaking early. pretty neutral matchups on gyms, and two of the E4 resist it, but Fairy is an OP type, and you should be fine.
OrAs: Eevee is dexnav only, and there's pretty much no postgame here. I can't really recommend anything here. Sylveon, I guess, looking at the movepools.
SM: While it looks like either Esperon of Vap.. What's this? an early game work up tm? that can be learned by a bulky Pokemon? that's right, Umbreon is finally viable. Now with work up, moonlight, and any dark move Umbreon can set up, and sweep. Solid matchups too!
UsUm: See SM. still early game. Still amazing. Even better now that Umbreon helps against ultra necrozma as a dark type.
The Lets go games: These games are so easy you could use any Eeveelution and get away with it. If playing Pikachu, use Vaporeon. Bulky, and can take the ground types that Pikachu can't. If playing Eevee, just stick with your partner Eevee, it's so OP you don't need an evolved Eevee.
SS: Eevee is one again available early game. Is Flareon finally viable? Superpower, Flare Blitz Work Up and Double edge? For the first time, Flareon my be the best for a playthrough... If it wasn't outclassed by Leafeon. Sunny day, Swords Dance and Solar blade are amazing, and Leafeon has enough physical bulk to set up on physical attackers. Good matchups as well.
BdSp: Guess what? no more Eevee in the trophy garden! If you want to use one in post, use Vaporeon, which now learns good moves way earlier then before. OP since gen 1 and counting.
LA: Eevee is found in the horseshoe plains. I don't know much about this game, but looking at learnsets, Espeon or Sylveon look the best.
SV: Is Flareon viable yet? (checks learnset) nope. Sylveon has good matchups against most bosses, just avoid revavrooms. Good movepool as well.
Z-A: Flareon is finally viable! Swords Dance, Flame Charge and Flare Blitz! Unfortunately for Flareon, Glaceon is actually really good here. Good Spa, and okay defenses. Speed is much less of an issue in Z-A, with the real time combat. Ice beam and Shadow ball hit from a distance, allowing Glaceon at act as a ranged fighter. Jolteon could work well here too.
So there you have it, the "best" Eeveelution for each main series game. Remember that much of this depends on your team. It's kind of a lengthy answer, I hope it was understandable. As I have not played all of these games take my word with a grain of salt.
Sources: Spending hours on the DB looking at learnsets, locations and gym leaders for every game, and experience from playing the gen 3 and 4 games for the gen 3 and 4 sections.
