How to legitimately catch a Mew on Red, Blue or Yellow:
Advance through the game until you reach Cerulean City and heal at the Pokémon Center. DO NOT enter the Gym at this stage. Battle your way through Nugget Bridge, and then find and catch an Abra without letting the Jr Trainer in the grass see you and trigger a battle. Now, determine exactly where the Jr Trainer is looking, and stand in the same column as him, but with him off screen. At this point, SAVE in case you mess up.
Now, move down one step so that the Jr Trainer will see you and trigger a battle, but as soon as you start moving that one game square into his line of sight, pause the game, and have Abra Teleport you back to Cerulean. If done correctly, the Teleport will work and the Jr Trainer will then have the "!" sign above his head, indicating a battle - which will not happen. After this point, pausing the game will not work.
Back in Cerulean, enter the Gym, and battle the first Swimmer. Once he has been defeated, return to Route 24 (the Pause menu should now work again), and continue to head north. If everything has been done correctly, the Pause menu will then activate on its own - and upon closing the menu...
N.B.: It is theoretically possible to pull this off using any Trainer that you have to walk towards in order to trigger a battle, but the Jr Trainer mentioned above is thought to be the easiest, and that the Swimmer is not a compulsory choice either. To that point, you will notice that this person doesn't even use the exact method I have suggested and instead uses the Youngster at the end of Route 25 (another common choice)! Also, note that this was not discovered until 2003, which is why so many people are unaware of it.
[YOUTUBE]J6hU-f_Q8Vk[/YOUTUBE]