What is interesting topic for research?

Different people will find different topics "interesting," but here are some ideas:
(1) Biographies: There are a large number of interesting people who have contributed to mathematics. There are towering geniuses like Euler, Gauss, and Ramanajuan. There are huge intellects that have made enormous contributions like Euclid, Appolonius, Newton, Liebnitz, various Bernoullis, and Descartes. There are people with interesting stories who made important contributions such as Napier, Noether, Hypatia, Ada Lovelace, Lewis Carroll, etc. Then there are the many non-Western contributors who receive little attention in introductory texts like Shang Gao (Pythagorean theorem) and Omar Kyyam (algebra.)
(2) Specific theorems: Many books have been written about the "Pythagorean Theorem" including hundreds of different proofs by multiple authors throughout many cultures and time periods. There are the fundamental theorems of arithmetic, algebra, and calculus; each with interesting histories and people associated with them. There is the "pons asinorum" of Euclid's elements.
(3) There are interesting problems, some of which are still open. The three problems of antiquity (squaring the circle, trisecting an angle, and duplicating the cube) all have vast historical attempts at solutions -- the ultimate solution requiring new mathematics in group theory. There are magic squares (another topic independently studied across cultures) and other arithmetic puzzles including Goldbach's conjecture.
(4) Another source of research topics would be branches of mathematics and the problems within them: number theory, topology, algebraic geometry, analysis, etc. all provide a rich area for research.