Eevee supposedly evolve to best survive/take advantage of it's surroundings. Evolutionary stones are essentially high doses of habitat, forcing the Pokemon to evolve to survive in its extreme surroundings. E.g., 'wild' Eevee might evolve in to Vaporeon if in an aquatic biome, or out of necessity if drowning, or through a stone. Umbreon might come about naturally if being attacked and severely damaged at night. Doesn't make as much sense for Espeon, Slyveon, etc, but you catch my drift.
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I don't think a Nomal-type Eeveelution would be viable in battle.
We had this discussion like five thousand times when folks were speculating about Sylveon possibly being Normal-type. Yes, mono-Normal is basically the worst typing that a Pokemon could have nowadays, but there are still ways to make it work, especially given the right combination of stats, movepool, and ability.