EarthBound

Giygas invasion!! Bizarre aliens roam the earth!!
Armed with a cracked baseball bat, a cookie and an ATM card, a courageous boy named Ness vows to save the planet from hideous destruction. He must first find his heroic companions, Paula, Jeff, and Poo, along the way to eight mysterious places called "Your Sanctuary" in order to gain power and put an end to the interstellar threat. The citizens of Onett lend their support by providing high-tech gadgets, robotic time machines, and wacky stories about how life has become.
Lurking aliens and inexplicable phenomenon are being reported throughout the countryside. Our four wonderkids band together with innovation, psychic powers and resourceful weapons. Will Ness succeed in his attempts to ambush countless frightening monsters as he travels? The ultimate test lies in the impending encounter with the all powerful Giygas!

EarthBound (Mother 2 in Japan) was the first quirky Earthbound-inspired indie RPG. The game is about four children who have to go on a quest to defeat an alien eldritch horror named Gigyas.

There's a wiki for it here, and a TVtropes page for it here.

EarthBound Series

Technically, EarthBound is the second in a series of games, but you can play any of the games in any order without being confused.

Games:

Fangames:

Playthroughs