I'd watch for loss of appetite, vomiting, lethargy, and general change in overall attitude and normal behavior. If you notice any of these signs, he needs to see a vet immediately.
If this were me, I would take him to a vet first thing in the morning reguardless. I have had a dog swallow an object it wasn't supposed to (foam that it tore off from somewhere) and I didn't wait for signs to show up. I took him to a vet immediately and had x-rays done to make sure it wasn't causing a blockage.
The thing with those rope toys is that they can be dangerous if ingested and the string can wrap around and cause some serious internal problems. I'd take him to a vet just to be safe, if you are sure that he ingested some of the string.