Yes, it's true: Surfing Pikachu is hidden in Pokémon Stadium. To teach 
your Pikachu Surf (a technique it can normally not learn), you need to 
first unlock the R-2 (Round 2) mode. Then do the following: 

Enter the R-2 Prime Cup's Master Ball division (you have to beat the 
first three divisions first). Assemble a team that contains a Pikachu 
that does not yet know Surf. You HAVE to use a Pikachu from your Game 
Boy game. You cannot unlock Surfing Pikachu using a Rental Pokémon. 
Do not register your team! Pick it directly from your Game Boy cartridge. 

You now have to beat the Prime Cup's Master Ball division using a three-
Pokémon team that always contains Pikachu. Pikachu has to be in every
 one of the eight battles. You are not allowed to use any continues. 
If you do, you won't get Surfing Pikachu.

Once you have fulfilled all the above conditions and beaten the cup, 
you are shown an award screen with Pikachu standing on a surfboard. 
From here on, Pikachu can use the Surf technique both in battle and 
during gameplay (Game Boy versions only). If you've got Pokémon Yellow,
you can now also access the hidden Surfing Pikachu mini-game. Simply 
go to the small hut on the northern beach of Route 19.