Which is the correct noun that should be used?

The correct answer is reason, because of the phrases "unexplained bruises" and "for no apparent" and "may indicate". These are all hedge phrases. If cause were the answer the sentence would read something like. "Where no cause can be determined/found in the case of bruises, this may indicate a bleeding disorder.
By the way, bruises are found on, not in, old people....

'reason' is the best choice, but cause could be accepted.


for reason

because of no apparent reason