Certainly, you can do that! Tying will be easiest, and can be done so the quilt still looks good. If the quilt was made of pre-washed fabric or has been washed, you should wash your new backing, of course. Cut it larger than the quilt -- about 4 to 5" larger all the way around. Spread the quilt face-down on a large flat surface, and then spread the new backing fabric over the quilt. Beginning in the center of the quilt and working toward the edges, safety-pin or straight-pin all over the quilt at fairly short intervals. If you use straight pins, then baste all over at fairly short intervals and remove the pins. Now turn the quilt over. Choosing places where the back side of ties won't look bad or may even add some interest to the top of the quilt, take a brightly colored thread that contrasts with the new backing and add temporary ties in all those places, but tying them very loosely so they can be removed easily. Now flip the quilt over again. Every place you have a temporary tie, add the permanent tie with the material (yarn, floss, etc.) you have chosen. Once the permanent ties are in place, snip the temporary ties to remove them, and remove all of the safety pins or basting. Now all you need to do is finish the edges! You can trim carefully, tuck the edges of the new backing under, and blind stitch the edges at the binding, or add a new binding. If it really, really matters, you can unstitch the original binding, trim and tuck the new backing under the original binding, and re-stitch it. Your quilt will now have a neat backing, be tied so it won't shift but with ties that don't make the front look wonky, and have that one extra layer of fabric that makes it just a wee bit warmer.