Earthbound Beginnings/WMG

Snowman is probably the MOTHER equivalent of Canada.
Its up north, always snow...

Giygas isn't just any female. Giygas is Maria.
Maria and George were abducted by aliens planning to run some experiments on humans. George managed to escape from them after some experiments and stole important information about PSI. Maria didn't manage to escape, however. The aliens mentally and physically altered her. She was turned into a shriveled, pale humanoid, and was also brainwashed by these aliens, although she eventually developed fond memories of her former self.
 * As for George, he didn't fully recover from his mental alterations. From the moment the aliens screwed him up, he started seeing things that weren't there. Things like Maria taking care of a small alien, or huge robots towering over him (He recorded the latter visions to build EVE). He also studied what he could about PSI, which was then learned by Ninten and Ana, and possibly passed down to Ness, Paula and/or Poo. Before long, his insane hallucinations grew too strong, and he eventually died as an old, lonely and insane human being.
 * Maria was locked in a capsule and taught many of the aliens' secrets, including PSI. She was now one of them. But before they could do anything else, Maria (now being known as Giygas) started unleashing an inexplicable power (PSI), and proclaimed herself the ruler of the aliens. Due to Giygas' incomprehensibly mighty power, they had no choice but to comply. She then ordered them to invade Earth, and the events of Earthbound Zero began.
 * After the events of the first game, Giygas vowed revenge upon the humans who defeated her, including her own great-grandson, Ninten, and fled back to space. She stayed in her vessel orbiting the Earth. When Ninten started singing the 8 Melodies, something cracked in her mind. It made her snap, and her alien mind took over most of her former memories. Her mind ripped apart from her body. The vessel eventually crash-landed on Earth, on a small town called Onett. There, she brought chaos and darkness upon the whole world, until Buzz Buzz decided to go back in time and find Ninten's descendant. Buzz Buzz was one of the aliens under Giygas' clutches, who went to Earth after notice of how her vessel had crashed there. While Buzz Buzz was searching for clues inside the crashe vessel, he heard voices outside. He found a boy who surprisingly looked a lot like Ninten, and assumed that this was the kid he was looking for. He then warned, the boy, Ness, of the horrible future that awaited the universe if he didn't stop her at all costs. This caused a rift in the space-time continuum, and Giygas, now just a mind existing without a body, was trapped by it in a land where time can't exist, inside a demonic capsule called the Devil's Machine.
 * After many adventures, Ness and company reached this place and encountered Giygas. Porky claimed that he was working with Giygas, even though he didn't know of her existance until she started influencing the minds of animals and weak humans. He just did that because he was a massive jerk, ready to build the most ridiculous plans and do the most vile actions just to mess with Ness. Thus, Giygas was freed from the Devil's Machine by Porky, ready to finally unleash her revenge on who she thought was Ninten. She was stopped, however, by Paula's praying. This insignificant soul was blocking her thoughts. Why? Because...

Paula is a descendant of Ninten, Ana or Lloyd, and she saved the world by breaking the limit of her PSI.
Paula was the only one who could put a stop to Giygas' madness because she was a descendant of the boys who put a stop to Giygas before. She prayed for someone to help them, and Giygas caught whiff of Paula's familiar thoughts. This started breaking down Giygas' mind even further, warping into unintelligible forms and corrupting itself. Giygas did one last attempt to stop them by devouring Paula's thoughts. Paula panicked, and wished for anyone, anyone to help her. Her PSI powers powered a plea was so strong that it warped through space, time, dimensions and reality itself, until it reached the player. This incredible breach between reality couldn't be fathomed by Giygas, and her mind devoured itself to stop this agonizing realization, erasing the chunk of time and space where the Chosen Four were, and their souls went back to their bodies.

Giygas's species is Deoxys from Pokemon
Eve looks a lot like Deoxys, and it's likely that Eve was made by Giygas's people, and was some of the technology that George stole. It makes sense for the race to have made robots in their own image, hence they are the same race as Deoxys. If you look at Giygas's sprite, it could be a really low resolution Deoxys.
 * Um, no. MOTHER came before Pokemon.
 * And who's to say MOTHER didn't simply introduce the species first?

Alternatively, Giygas is Mewtwo.
Mewtwo was explicitly designed to look like Giegue, IIRC. The guess follows.