I think something is worth studying if it helps one of:
solving a problem I know about,
giving a new perspective on something I know, or
raising interesting questions, some of which are easy to solve and some of which aren't.
Especially, I study it if it gives me some degree of gratification. Here are a couple of examples of things that I hope to pursue after my current interests wane:
Recursive clone theory: A class of functions on a set which is closed under composition and having projections is called a clone; the notion is a part of basic general algebra. Something that should be mentioned in basic recursion theory classes but is not is that various definitions are specializations of clones: primitive recursive functions, partial recursive functions, total recursive functions. I think it would be useful to blend the ongoing research in clone theory with a computational component that can answer how complex a class can be.
Transforming Shelah's classification theory: In determining how many inequivalent models of cardinality kappa exist for a first order theory, Saharon Shelah came up with conditions on the theory which (loosely and inaccurately speaking) sometimes dealt with whether a theory could encode a particular order or a certain simpler theory. I think the ideas can be moved into the domain of computation over finite structures. In particular, languages that are members of some complexity class (oh, say, NP) could be shown to satisify properties analogous to what Shelah developed for first order theories. I think that this would be a promising route to find a language in NP - P .
Granted, these are not generalizations so much as taking tools, trying them on a new kind of widget, and then retooling the tool to work on the widget. The justifications for working on them should be the same and (I think) apply to your questions.
Gerhard "Ask Me About System Design" Paseman, 2010.09.24