From my experience it is extremely hard to restore a crochet doily if it gets a hole in it, especially if you do not have the pattern used to make it. (I have made quite a number of doilies in my day and use them, so some of mine do get holes.) I sometimes just stitch with a needle and thread catching any loops of the crocheting and pulling them gently together to keep them from pulling out further and hide the hole the best I can. Then I continue to use the doily by setting something over the mended area. If you frame it, you could put several things in the frame, overlapping and covering the frayed part.