Here are my two cents.
EEVEE.potential - Currently has Fire, Water, Electric, Psychic, Dark, Ice, Grass, and Normal as possible outcomes. Following the pattern, Dragon would probably be one of, if not the only, added Eevolution. Meet: WYRMEON...
EEVEE.balance - Flareon beat Jolteon beat Vaporeon beat Flareon. Umbreon beat Espeon. Glaceon beat Leafeon. In 2 and 4, the easier one to obtain was beaten by the harder one to obtain. Here's the question: Would Wyrmeon be easy to obtain or hard? If easy to obtain, we simply give Glaceon a reason to be OU, or else create a Steel-type Eevolution: COPPEREON. If hard to obtain, make Flare, Jolt, Vap, and Leaf be more commonly seen.
EEVEE.novelty - How is the evolution of Coppereon and Wyrmeon to be handled? I think it's obvious - a item-trade evolution for both. We've already got the Metal Coat and the Dragon Fang established, it shouldn't be too big of a leap to see this working. HOWEVER: I would also like to see a bug Eevolution, SCORPEON. Give it a Toxic Orb before trading, and it comes out with the ability Poison Heal and heavy physical emphasis at the expense of special attack and defense. Since it's a Bug type instead of Poison, it can still damage Coppereon, but it still plants Coppereon firmly in the HOU category over Wyrmeon and itself.
EEVEE.statting - The only defensive Eevolution is Umbreon, but Coppereon would change that, being the Eevolution best suited for soaking up physical damage. Wyrmeon would probably end up as what Flareon could have been, a special sweeper with a penchant for fire-based moves. Scorpeon would probably need to take up the Physical Sweeper role, and a good movepool would make up for its ineffectiveness against Coppereon.
So, TL;DR: Bug and Dragon, and Steel.