To date, three stand out.
The first is Final Fantasy, since like the games it doesn't have to build directly upon the world established in any previous incarnation of the series but rather uses some form of shared lore (spell names, creatures, etc). While bringing such a world to the stage would be an expensive endeavor for something that at its cheapest would be just as expensive as most AAA games and at its most expensive would be 2-3 times as high, I think it'd be a spectacle worth watching, though even if it lasts only a year on the stage before closing, it'd have to make its money back through merchandise, cast recording sales and the home media release of any official pro-shot.
The second is Crash Bandicoot. Given how Julie Taymor has adapted the animal characters of The Lion King and how Tina Landau has adapted the anthropomorphic aquatic characters of SpongeBob Squarepants to the stage, I could see Crash Bandicoot done in a similar fashion. My top choices for the cast would be Billy Porter as Aku Aku, Sophia Caruso as Coco and either Ethan Slater or Andrew Keenan-Bolger as Crash Bandicoot.
The last is Super Mario, which given its appeal would be the most marketable option, though Nintendo's protection over its IPs might make such a project a fantasy at best. I could see Danny Skinner and Ethan Slater as Mario and Luigi, and given their roles as Patrick and SpongeBob in the SpongeBob musical they already had an excellent rapport with one another. To round out the cast I'd suggest Taylor Louderman as Peach, Alex Brightman as Wario, Wesley Taylor as Waluigi and either Will Roland or George Salazar as Toad.