The phrase "old conundrum" can refer to a perplexing problem or situation that has been around for a long time. Some synonyms for this phrase might include "age-old enigma," "timeless puzzle," or "enduring mystery." Other options include "perennial dilemma," "long-standing paradox," or "persistent riddle." Depending on the specific context, other words like "classic problem," "eternal conundrum," or "evergreen challenge" might also be appropriate. Whether referring to a philosophical debate or a practical issue, using these types of synonyms can add some variety and interest to your writing.