Instructions
Beat the game's main storyline to receive the National Pokedex upgrade.
Prepare a team to go catch Meowth. Put a Pokemon that can learn False Swipe, such as Kricketune or Absol, in your first slot and make sure to have a Pokemon that knows Hypnosis, like Drowzee or Gastly.
Stock up on Pokeballs at the nearest Poke Mart. Dusk Balls are the most effective if you're playing at night; otherwise, choose either Nest Balls or Ultra Balls.
Fly to Hearthome City, then head south to Route 212. Continue until you reach the Pokemon Mansion.
Enter the Pokemon Mansion and speak to the owner, Mr. Backlot. He will be in one of the rooms along the hallway to the right. By speaking to him, you will find out which Pokemon are available in the Trophy Garden behind his house that day. Meowth is one of 16 different Pokemon that may appear in the Trophy Garden. If Meowth is not among the Pokemon mentioned that day, try again.
Head out to the Trophy Garden if Mr. Backlot says he saw Meowth that day. Walk around in the tall grass of the Trophy Garden, entering battles with wild Pokemon until you encounter Meowth.
Attack Meowth with False Swipe, which should reduce its HP to one without knocking it out.
Switch Pokemon, sending out your Pokemon that knows Hypnosis.
Use Hypnosis on Meowth. If it fails, try again until Meowth falls asleep.
Throw Pokeballs at Meowth until you capture it. Catching Meowth should be easy with its HP so low and the sleep status will raise your chances even higher.