General:
* Mystery
* Puzzle
* Riddle
* Conundrum
* Paradox
* Cipher
* Secret
* Unsolved problem
More specific:
* If it refers to a person: mysterious figure, inscrutable person, perplexing character
* If it refers to a situation or event: baffling situation, inexplicable event, strange phenomenon
* If it refers to a concept or idea: unfathomable concept, inscrutable theory, incomprehensible thought
The best synonym for "enigma" will depend on the specific context in which you are using it.