At fabric stores, there is a product called grosgrain tape. I buy it in white and use it to mend frayed collars.
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If you can't find that the other product that I would use is bias tape.
Whichever tape you have would be pinned over the frayed area and stitched on both sides.
Depending on the fray the tape might need to be on both sides.
I am not sure how attractive this will be though it will work.