I can't believe I'm the first to say this, but I know how to get mew.
First, you can't have beaten the trainer with a slowpoke above nugget bridge, or the trainer with two poliwags and a poliwhirl below the underground tunnel entrance on the Lavender Town side.
Also make sure you also have a pokemon that knows fly and something that can weaken a L7 mew, mabye a pikacku L14 that can paralyze and damage mew without fainting it.
Go to the entrance above the 2nd trainer mentioned without him showing up on the screen and save.
Then take one step down and immediately press start and your menu should pop up; otherwise you battle the trainer.
Choose a pokemon to fly to Cerulean City. You should be able to fly, but a ! should still apper above the gambler's head.
Once you get to Cerulean, your start menu will stop working because the game thinks you're still in a battle.
Go to the trainer with the slowpoke and let him see you or else the game will lock down. Once you beat him, fly back to lavendar town and head west.
Your start menu should pop up on its own once you go far enough, so press B. You better have a lot of great balls because once you press B, here comes mew.
The down side is that mew only knows pound, but leans something at levels 10, 20, 30, and 40.