I have some of my grandmother's doilies also. I have glass on some of my furniture so instead of those little clear, plastic rounds, I use the doilies underneath the glass. It gives it the space that the rounds would, and I get to use and see the antique doilies. Because of being under glass, I very rarely have to clean. I think not cleaning often, preserves them.

If you mean the kind that have printed designs, such as large irises, and are usually in squares that would fit a bridge table, I have seen cute quilts made from them at a quilt show, and to my way of thinking, it's a wonderful way of getting them out of storage and out where they can be enjoyed. If you're not sure of the value of your tablecloth, you might be able to see something similar on eBay and check what the final bids have been on recent auctions (Asking price says nothing!). That would tell you whether you have a rarer one that probably should be preserved as is, but in any case, it's yours to do with whatever you want. I am using one as a window topper, but I folded it in half on the diagonal and draped it through scarf holders so that I didn't have to cut it. It has no particular sentimental value for me, however (bought at an "antiques and collectibles" store), so I would not hesitate to cut it if I wanted to make a bed runner. In fact you could cut one on the diagonal and use a compatible fabric to make alternating triangles that would go with it to make a longer runner to drape on each side of the bed. That way the whole tablecloth would show up. Use whatever fabric you like on the back and quilt it with a thin batting. Or you could quilt it to a layer of fleece with no batting, and the result would be a throw that drapes more softly.

I'm another of those who still uses tablecloths, but I don't presently have a cat! I like the idea of just folding one plain tablecloth over a layer of leftover batting, stitching it closed, adding enough quilting to keep batting in place, and giving it to the cat!