Alternate Pokemon Colors
Did you ever notice how some of the Pokémon that appear in Pokémon Stadium have different colors? The secret to the color variations lie in the names you give your Pokémon as well as your trainer ID number (assiged to you in Pokemon Blue, Red or Yellow, whenever you start a new game). While there are a few secret names that change only certain Pokemon's colors, there are a few naming tricks that result in different color variations. For example:

Use only lower caps (ie: poliwhirl)
Reverse the syllables (ie: DRILLBEE)
Abridge the names (ie: Pika)
For a more detailed listing of color differences, keep an eye out for IGNGuide's Pokemon Stadium Guide.

Open Up R-2 Mode
To get the secret R-2 Mode (as well as a brand-new title screen) defeat Mewtwo after you have beaten the Stadium Mode and Gym Leader Castle Mode. You will now see the final credits sequence, which will unlock the harder R-2 Mode and all new battles. You can toggle between R-2 and Regular Mode at the start-up screen by pressing C-Right.

Win Bonus Pokémon
Each time you beat the Gym Leader Castle (including the Final Four and your Rival), you will receive one of eight semi-rare to rare Pokémon. These Pokémon are awarded at random, so it's likely that you'll need to beat the Gym Leader Castle again and again to get the ones you want. Note that once you have beaten the Castle, you only need to defeat the Final Four and your Rival again if you want to win more Pokémon.
Here are the Pokémon you can win:

#001 Bulbasaur
#004 Charmander
#007 Squirtle
#106 Hitmonlee
#107 Hitmonchan
#133 Eevee
#138 Omanyte
#140 Kabuto

Rent Mew
Many of you have asked how you can get all 151 trophies in the Hall of Fame if you don't own the elusive bonus Pokemon Mew. No problem: Once you open up R-2 Mode, you can rent and use Mew in the Prime Cup (sorry -- you still can't get Mew in Gym Leader Castle and Free Battle).

Battle Mewtwo
To unlock a hidden 6-on-1 battle with Mewtwo, beat the Gym Leader Castle and get trophies in all 10 tournaments in the Stadium Mode. The map screen will now turn to a night scene and a bonus tournament will open up in which you battle against the psychic super-Pokemon, Mewtwo (#150). Once you beat Mewtwo, you unlock a new game mode.

Bonus Stickers
Once you have beaten both the R-1 mode (including Mewtwo) and the R-2 Mode (including Mewtwo), you are given a set of bonus stickers that you can access in the Gallery Mode. When you're on the Gallery main screen, press and hold L and R and move the Cursor to Print. If you keep the buttons pressed, the Print option will now change to Bonus. You will now have access to a set of 16 bonus "wallpaper" stickers from all the different modes that you can print out at "Snap" Stations.

New Title Screen
Beat both the Gym Leader Castle and Stadium modes to change the title screen (in addition to the map mode). The new title shows depict Pikachu, Blastoise, Charizard, Venusaur and Mewtwo.

GB Speed Upgrades
There are two secret speed upgrades to the GB Tower Game Boy emulator that let you play Pokemon Red, Blue and Yellow at much higher speeds:

Doduo Mode: Beat the Poké Cup. Once you have won all the Poké Cup tournaments, the GB Tower changes into Doduo GB Tower and a Doduo sticker appears on the Game Boy. The Doduo upgrade doubles the speed of the game play when playing the Game Boy game on your N64.
Dodrio Mode: Play the Game Boy games at hyper speed with this alternate speed settings. To get it, you need to beat the Prime Cup (after you have already beaten the Poké Cup).
You can switch between the different GB Tower speed settings during gameplay by pressing C Right.

Get Surfing Pikachu
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.

Talking Pikachu
For a talking Pikachu, you need to upload one from Pokemon Yellow. Rental, Red, and Blue versions will just be silent.

Kids Club: Hyper CPU Difficulty
To unlock this hidden difficulty setting, select a single-player game against three CPU-controlled opponents. You need to win the "Who's the Best?" challenge five times in a row on HARD difficulty. This will open up the new HYPER difficulty setting for the CPU players.

Get Amnesia Psyduck
Have you ever wondered how you can get the elusive Amnesia technique? Well, it's hidden deep in Pokemon Stadium. Unlocking it will take a while, but it's definitely worth it. You have to register ALL 151 Pokemon in the Hall of Fame. Whenever you beat a tournament (or Gym Leader Castle), your six Pokemon are entered into the Hall of Fame. It doesn't matter whether you use rental Pokemon or your own, but you can't have any gaps. Once you've filled it all up, you are awarded a free Psyduck bonus Pokemon that knows the Amnesia technique.