Instructions
Restart your game from the beginning. If you've been playing for a while there is a good chance you've already done something to prevent you from getting Mew. Start playing from the beginning and continue until you get the HM02 Fly ability from the house that is west of Celadon City.
Avoid the Youngster that is on Route 25, the road that leads to Bill's house. The Youngster has a Level 17 Slowpoke. After crossing Nugget Bridge, the Youngster is the fourth trainer that you come across. He will be standing directly above a girl trainer, and will be facing north. Also avoid the Gambler that is on Route 8, the road between Saffron City and Lavender Town. He has 2 Poliwags, both of which are Level 22. He is east of the entryway to Saffron City. He will be facing north, looking towards the door that leads to the Underground Path. You must avoid these two trainers.
Obtain a Pokemon that has the Fly ability. While on Route 8, stand directly in front of the entrance to the Underground Path. You will want to be against the house. Save your game because if you try to save after starting this process, it will disrupt your attempt to get Mew. Take one step down and press "Start" immediately. If you do this successfully, your menu will come up before the Gambler notices you and engages you. Go to your Pokemon list and choose to fly to Cerulean City.
Go north and cross the Nugget Bridge. Afterward, go east until you come across the Youngster described in Step 2. Your "Start" button should not function at this point. Walk around the Youngster. You cannot walk right up to him or else your game will lock up. Allow the Youngster to notice you and approach you. Fight him and defeat the Slowpoke he battles with. Your "Start" button should function again after this. Press "Start" and fly to Lavender Town.
Head west after landing in Lavender Town. Head into Route 8. Once you enter a narrow path, your "Start" menu will come up all by itself. Hit "B" to exit the menu. Once you exit the menu, you will be in a battle with Mew. He is only a Level 7 and his only attack will be pound. Catch him with a Pokeball.