There is 2 ways to catch a wild Mew, That I know about... That's right 2. You can do them both to have 2 Mews if you would like. One of them is right above Cerulean City. You need to defeat the trainers and your rival on nugget bridge. Take a left at the top of the bridge to find the grassy area to your right, but be careful. Do not fight the trainer in the grass! Instead, go around him, and find yourself a wild abra. At this point in the game you should have a Clafairy or this could be difficult. Your Clafairy should know sing (level 13) and you need to put Abra to sleep, because Abra enjoys teleporting away when you want to fight him. Sleep him, and catch him. Now, head back to the Poke`Center in town to add Abra into your party. Head back up nugget bridge. Now, go left and again, be weary of the trainer in the grass... The easiest way to do this is to go to his left, and walk UP untill he is one space below where your screen shows. Take a step to the right. Save your game. Take one step down and IMMIDIATLY push the start button, select "Pokemon" And have Abra teleport you to Cerulean. If done right, you should Teleport away, just as the (!) Comes up, when the trainer notices you. Now take a right, to head into Misty's gym. Fight the swimmer in there, and defeat her. Now head out of the gym, and up nugget bridge. Your start menu should appear, So save your game when it pops up, cause once you hit the "B" button, you are going to be in a pokemon fight, with Mew.
And the other comes later in the Game, However, make sure you have not faught "Youngster" With the slowpoke, above Cerulean on the way to Bill's house. Head West from Lavender town, untill you reach the house that leads into the Underground Passage. There should be a Gambler around, make sure you do not fight him. Instead do the same thing as before, pressing start Immidiatly after you walk down into his line of sight, only instead of Teleport, you should have a bird who knows fly by now, so fly to Cerulean. He should get the (!) Above his head as you fly away. So go north from Cerulean, up the nugget bridge and to your right. Fight the Youngster, and then fly back to Lavender. Go the the West, out of town, your menu will pop up. Press "B" and fight a Wild Mew.
There are more trainers you can take on to catch various wild pokemon. Fly away from the Gambler, To any town with unfaught trainers nearby, fight them, then fly back to the area that the gambler was at. I have done this cheat to find many pokemon, some which i could not have caught in Blue alone. Try it yourself to find out diffrent trainers that give you diffrent wild pokemon. The secret to all of this is the game glitches when you should be fighting but don't. It is glitched the second you fly away. And when you fight a trainer, the Special power of that pokemon is what triggers the wild ones. When you go back to where you were, it recognizes a fight, but no wild pokemon, so it matches your special with that of another one, and whala! Pokemon battle.