I am interested in looking at "resolutions" of modules in noncommutative rings, making some obvious necessary modifications to the definition of a resolution. However, whenever I try to search the literature, all I find is resolutions of singularities in noncommutative settings. I don't know enough about geometry to know if these are related to what I want to do, but I suspect that they are not.
Is there just a paucity of material, or should I be using different keywords? Suggestions?
The reason your search isn't successful is that when people talk about projective resolutions or free resolutions, there's not usually an implication that they are dealing with modules over a commutative ring. Standard homological textbooks develop homological algebra in more general categories, including the category of modules over a noncommutative ring. The term "noncommutative resolution" isn't used for a free/projective resolution of a module over a noncommutative ring, rather it belongs to noncommutative algebraic geometry where one tries to generalize concepts like resolution of singularities.
If you want to know about homological algebra for not necessarily commutative rings, standard modern references are Weibel's An introduction to homological algebra or Gelfand and Manin Methods of homological algebra and Homological algebra. Classical texts are Cartan and Eilenberg's Homological algebra or Hilton and Stammbach's A course in homological algebra.