♦ ♦ ♦ Symbolic Appeal
Eevee and its subsequent forms are fairly popular among Pokemon collectors, so much so that the evolved versions have been dubbed "Eeveelutions." This amount of fan affection for a pokemon family is suprising, considering that eevee is not a player starter pokemon in the major video games. I believe the appeal Eevee and his eeveelutions are that they represent the freedom of choice that all people desire.
Often in our lives, humans live their lives feeling conflicted by various influences, such as family, religion, societal norms, etc, that push us into certain directions. These influences are factors in many of our decisions, from the small ones like how to dress, to the big ones like what career path to take. As a child, we are told that "you can be anything you want to be," but in adulthood we quickly learn that extreme idealism not true. Thus, people love escapism. Video games are a particularly notable form of escapism because you have this uncanny freedom in your decisions that are not limited by outside forces or even time.
Thus, getting back to Pokemon, we just adore Eevee and this freedom to choose where your pet is heading. With an Eevee, you're not confined to one set path or one set expression of yourself. If you like fire, make a Flareon. If you like defense, make an Umbreon. You have complete liberty to do what suits your unique interests. Eeveelutions recognize you, the pokemon trainer, as a individual with preferences that aren't always uniform with everyone else in the world. Who doesn't like customizable things?