Mew

Mew is regarded as the first PokéGod. It is a real Pokémon, but due to its rareness, fans believed that it must be somewhere in the game.

Origin
When fans learned about Mew, it was deemed the secret, super-powerful Pokémon that had a number greater than 150. Imaginations wandered, thinking of ways to obtain this mysterious Pokémon. The only official way at that time was to win it in a Nintendo contest, but fans explored to think of other ways to find Mew. Although none of these worked, fans knew Mew existed somewhere and it caused fans to think if there were more secret Pokémon, which is when the other PokéGods were created. Mew had such a big impact that Celebi was later used in a similar way for Pokémon Crystal, but no Pokémon had as big of an impact as Mew.

Location
Mew is thought to be found in a few different locations, as shown below.

Vermilion City
In Vermilion City's harbor, a pickup truck can be found. Once the S.S. Anne leaves the harbor, so does this truck. If you trade to get a Pokémon with Cut, you don't need to go on the S.S. Anne, thus keeping the truck in the harbor. Fans believed that if you came back later and used Strength on the truck, it could be moved to reveal Mew. This was soon proven wrong and the truck serves no real purpose in the game besides this rumor.

Indigo Plateau
Like many PokéGods, Mew was thought to be obtained after defeating the Elite Four a number of times. This was usually just a step in the process of getting Mew and there would usually be a high number of times and other requirements involved to make it near impossible.

Hall of Fame
The Diploma was an item obtained after completing the Pokédex, but it was rumored that Mew could be obtained instead with certain other requirements, including going to the Hall of Fame. Usually, the requirement was to have the Japanese version, which was considered to be mysterious at the time, but in the English version, it was thought to be obtained after completing the Pokédex. This was proven wrong fast, leaving fans to specualte more about how to obtain Mew.