What would be a good subject for a research paper on The Mambo Kings?

I assume you're referring to the book The Mambo
Kings Play Songs of Love by Oscar Hijuelos rather than the movie or some
other adaptation of it.  The movie does capture the essence of the story; however, some
elements just aren't the same, particularly in terms of scope. You also don't say if
you'll be writing an informative or persuasive essay, and that makes a difference in the
kinds of topics you should consider.  So, assuming we're talking about the novel, here
are a few ideas to consider:

• Compare and
  contrast Cesar and Nestor


• Compare and contrast Delores
  and Maria


• Examine the pitfalls of fame as demonstrated in
  the brothers' lives


• Explore the consequences, good and
  bad, of fame in their lives


• Examine the cause(s) of
  Cesar's descent into depravity


• Discuss the impact of
  other artists on Cesar's and Nestor's career and
  lives


• Decide how many of their troubles are because of
  their successes and how many are because of their
  failures


• Study Delores as a suitable match for each
  brother


• Examine the famously re-written song to Maria and
  how it reflects the changes in Nestor's
  life.

There are an unlimited number of other
choices, as well.  In general terms, remember a thesis must be provable or at least
something you can defend.  You'll want to be sure to choose something you feel strongly
about or that you believe, and you'll need to add your own perspective to it, as well. 
Make a list of ideas from which to draw, then choose the one you feel is the strongest
for you.  Support that with evidence from the text, and you'll be ready to
write.