About the second question: I think senior mathematicians don't necessarily escape the criterion of general interest, but it can become a self-fulfilling prophecy: The mere fact that a senior mathematician is studying something can raise interest in the object of study among the mathematical community - I guess they easier grant him that he will see connections or analogies to other areas accepted as interesting. See Minhyong Kim's nice "money in the bank" comparison.
About the first: Of course you want to study this concept you are interested in. So to make it interesting for others you could go for some introspection - what is it that you find intriguing about it? Can you pass it on to others (this is surely easier in talks than in papers)?
It does not always have to be a big range examples that apply to it. Maybe you feel it behaves unexpectedly well in spite of weak axioms. Maybe it clarifies that many of the facts about Y depend only on the fact that it is an X and thus improves the understanding of the well-accepted theory of Y. Maybe you have a single application where it showed up and feel that there it greatly helped to separate the algebraic content of the situation (which is strictly more than the structure of a Y) from the rest. These seem all like potential good reasons to work on the theory of X.
But maybe your fascination comes from the feeling that your X shows unusual behaviour for an algebraic structure, then spelling that out you could find that this just reflects your prejudices about algebraic structures, which others don't have - this could be a criterion record this as learning experience and do something else for publishing...