Instructions
Give your Eevee haircuts in the Goldenrod Underground. Save the game in front of the haircut desk and restart the game after each haircut. Do this until Eevee has gotten the maximum amount of happiness points from the location.
Fight other Pokemon with your Eevee. Set her as the first to fight to maximize the amount of happiness points she will receive. Use the "Sand-Attack" move with Eevee during a fight to lessen the chance of your opponent hitting you, thus letting you fight more powerful Pokemon.
Wait until the time in the game is between 4 a.m. and 6 p.m. and Eevee is close to leveling up.
Fight wild Pokemon or trainers as early in the day as possible to ensure your Eevee will level up during the day and evolve into an Espeon. Eevee's level does not matter when you're evolving it into an Espeon -- only its happiness points matter. If it has 220 happiness points or more and it is daytime in the game, this Pokemon will evolve into an Espeon.