Well, there is no "best" HA in terms of statistics, rather much more opinion based. Therefore, to find the best HA eeveelution in your view, compare how well it would work in your team.
Flareon (Guts): Guts is used to boosts the ability-bearer's attack by 50% when inflicted with a status condtion. This ability is not really controlled by you, unless Flareon is holding a Toxic/Flame Orb, or have an ally inflict one on you. However, having an ally using it on you can be a waste of a turn in a crucial battle, so have hold either a Flame/Toxic Orb. A Flame Orb would not be the best choice since a burned Pokemon will have its attack lowered; the exact opposite of what Guts is trying to increase. A Toxic Orb is very popular, since you only need to worry about your health, however it does badly poison you. Another choice that is not as common is having an ally paralyze you. Flareon's speed is already slow, so you should pretty much consider yourself going last - however you do need to worry about 25% chance of losing your turn. So, a Guts Flareon is very questionable to use.
Vaporeon (Hydration): This ability will cure status conditions inflicted to the ability-bearer. However, you will need Rain to be in play in the battle. Therefore, either have Vaporeon learn Rain Dance, or have an ally with Drizzle. Hydration can also cure self-inflicted status condition, so a move like Rest is a good idea to have it learn - but you would need to take Vaporeon's low Def into consideration.
Jolteon (Quick Feet): Similar to Guts, it raises the ability-bearer's speed when inflicted by a status condtion. So, the same ideas used for Flareon can be considered, except paralysis, as Jolteon is an electric type. The best way to take advantage of this is holding a Flame Orb, since a burned Pokemon lowers attack, but Jolteon is much better in SpAtk. Jolteon is already one of the fastest Pokemon on a field, but Quick Feet can almost ensure you always going first.
Espeon (Magic Bounce): Magic Bounce reflects status-changing moves, including status conditions and entry hazards. This ability is very useful and powerful, but it is almost out of your hands in terms of activation.
Umbreon (Inner Focus): This prevents the Pokemon from flinching. Umbreon already has great defenses, so it should be able to withstand an attack, whether or not it causes flinching. The only super-effective flinching moves would be Low Kick, Rolling Kick, and Steamroller - and these moves aren't very common. You would usually find this ability being useful when against popular flinching moves such as Fake Out or Rock Slide.
Leafeon (Chlorophyll): This doubles the ability-bearer's speed in sunlight. So, you should consider having Leafeon learning Sunny Day, or have an ally with the ability Drought. Leafeon is definitely one of the fastest eeveelutions, so this speed should be beneficial, not only for you going first, but to have a higher chance to attack before a super-effective move lands on Leafeon, as it does have a lot being a Grass Type.
Glaceon (Ice Body): This ability heals the Pokemon during Hail. Consider having Glaceon learning Hail, or have an ally with the ability Snow Warning. You can have the Pokemon hold Leftovers or another healing item to increase HP per turn - unless you lose the item by a move such as Knock Off.
Sylveon (Pixilate): This changes Normal moves into Fairy moves. Sylveon does learn Normal moves that are stronger than it's Fairy Moves, such as Hyper Beam - and Return might be a good move since Syvleon is based off of affection. Also, Fairy typing does provide weaknesses to other types in which Normal moves never had. These type of abilities are always a good choice since it differently opens up your pool of moves to select from.
Overall, all abilities are good and can be used for different situations. If I were to give some suggestions, Espeon and Sylveon are some of the most popular and effective in my opinion, and I think it will really help you. However, as you know, it's ultimately up to you to decide.
Your Choice! Hope I helped! :)