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