So when the Xbox 360 was still "new" I'd say like around 2007ish/8ish when the first Call of Duty games were coming out.
There was a game that I remember playing but I cannot remember its name. It was sort of an action/adventure game that had a girl sort like like Lara Croft style and she could turn into a blue "monster" like James Sullivan from monsters inc.
Most likely Kameo, but that "Lara Croft" reference... I don't know, the only thing that comes to mind on the 360 when you say: shapeshifting female blue monster, is as Ray said; Kameo. If we're wrong I'd love to hear more about this game in order to help you find it.