Sorry, not really.

You can disconnect from the network (file transfers can still continue when disconnected) and minimize LW. That way your search tabs will remain up, and you could just use the Repeat Search after reconnecting.

Or, you can cut and paste search terms into another application.

btw--saving search results wouldn't help much--they get stale pretty fast since quite a few hosts just jump on the network for a bit then jump off. A Repeat Search would be needed to find fresh hosts anyway.