It looks like you have some applique on this quilt? Any chance you could work a bit of applique so that it covers the frayed area? Another idea would be to make this an opportunity to design a neat tag with some custom embroidery or layered ribbon in coordinating colors to the quilt -- store bought items have tags in prominent places, so yours can too. Make it look like you meant it. Those would be my first ideas for a fix.
I hate ripping bindings off, specially after a quilt has been washed. I think the risk of it not looking not quite right after re-binding is fairly high, so if neither of the suggestions above works, I would probably carefully pull the quilt top material to toward the binding, possibly use fusible web to hold it, and then gently turn under the frayed area and take tiny stitches to almost invisibly tuck it under neatly. If I was still worried about fraying, I would use matching thread and do some interesting embroidery, like an area of cross stitche or blanket stitch.