I haven't read many superpower books, but let me list a few ideas that may not be cliché.
1) Physical or emotional trauma. Make a situation where the character needs the power to survive or help others, but they do not have it. The power is unlocked, unbeknownst to the character until another event happens where they can use the power. This has been done, but there are many mental and physical illnesses that you can choose a rare one to unlock the superpower. Phobias could help place pressure on the mind to bring out powers. Just like how someone can be triggered by forgotten traumas, the superpowers can come like the memories. (Might have to mix in psychology which could be difficult to explain logically.)
2) This one might be very cliché- Magical objects that grant them powers? Perhaps a random stinky shoe they found on the road or a strange amulet.
3) If there are others in the world who have superpowers, maybe have the character touch someone with powers and some of their ability will be passed onto the character.
4) If the novel mentions deities or superior beings, maybe have them grant the powers? This one is quite cliché in cultivation books though.
5) The character finds a phrase or book that lets their powers flow after being blocked for so long. An ancient chant is a cool example. Actually, this might be cliché in fantasy novels. I don't know.
6) Losing a limb that prevented the power from activating? I can't imagine how to explain that though.
7) They underwent a surgery where they received an organ from a donor who had superpowers. They have a piece of the person's power. (Not supposed to be scientific at all.)
8) Hearing a song. The song could be like a call bringing forth the powers.
9) Simple abilities evolve to become superpowers. For example, someone can yell very loud without superpowers. Then, the ability becomes deadly screeches.
10) The character searches for their abilities for years until they finally say "Damn, I wish I had superpowers!" A star shoots across the night sky and bam! Their powers have arrived.
11) They do something stupid like eat a rock.
12) Gaining superpowers in steps. Use a few of the ways a superpower can be gained and combine them to lead up to the final product. One idea unlocks the power. The other helps it grow.
That was interesting! So many things have been done that this list was so difficult to make and even then, half of these have been used in some way. Many of them are not logical, which is also a bit disappointing. Anyway, good luck with your writing! Sorry for writing too much.