Short Version: In addition to the normal 3 evolutions, Eevee could evolve into a hole spectrum of different Pokémon types
Why We Believed It: The Pokédex data suggested it was possible and given the variety of element stones in the game, it seemed rational
In the original Red, Blue, and Yellow editions of the games, Eevee was the only Pokémon with a branching evolution path. Using either a Water, Thunder, or Fire Stone on its basic form would transform the fluffy little creature into a more advanced, elementally charged version of the species. Naturally with three of the five elemental stones in the games working, folks began to wonder if the Moon Stone and Leaf Stone could be applied to find new, rare Pokémon.
It all made a lot of sense given Eevee's Pokédex data which read, "Its genetic code is irregular. It may mutate if it is exposed to radiation from element stones." which for many implied that it will mutate if exposed to element stones. Eventually folks smartened up and realized that it would be illogical for Vaporeon, Jolteon, and Flareon to be immediately adjacent entries in the Pokédex and then a grass-type to follow after #150.
Still, Nintendo knew folks loved finding new Eevee foms and included four more (including a grass-type, Leafeon) in subsequent releases. Still, that didn't stop young players from coming up with wild theories on how to find even more unique types of Eeveelutions.