Finding it hard to find a Master's thesis advisor in mathematics, all the professors I've asked say they are too busy I attend a competitive mathematics department that is very active in research.  The downside of this is that it is hard to get any professor to agree to work on a Master's thesis with.

I have tried asking about 5 professors - all say the same thing, "thank you for your interest in wanting to work with me.  I'm afraid I have to say no, because I am very busy."

To be honest, though, I did ask the top researchers in our dept.  Partly because I should take that risk for my own career benefits, and partly because I just do not know much about the younger, less established professors / post-docs, people whom I am guessing I would have a better chance to work with.

What are some general recommendations from this community?  Keep trying to work with a tenured, well-known professor, or better to hedge my bets and just find someone who will say "yes, you can work on a project with me."?
In your situation, one might advise that you better forget about the elite faculty members, now, and the mere struggle to work with them. There are two potential options in my mind.

Firstly, expand your professor set to young fellows and try to grab their attention. You will be able to flourish by the indirect guidance of the elite professors, as your co-advisor; but right now, delay within pursuing the thesis might be considerably harmful for you.

Second idea would be taken into account, if even young and inexperienced fellows at the department, corresponding to your desired research field, show no tendency for your supervision. In this case, you may have to switch from your current research interest, in an either partially or fully matter, to be capable of cooperating with the people, who are looking for any graduate student.