I use 5mm or 6mm line. If it needs to break, I hope it does. If it doesn't need to break, it keeps the oar floating near the boat and easy to pull back aboard.
I just tie a bowline around the oar shaft barely loose enough that it can slide freely. I have a Cambridge frame, and I tie the other end onto the oar tower with a modified bowline. I don't pull the tag end all the way through, but just tighten it up on a bight. If I need to release it, I just tug on the tag end, and the bowline will release.
And since nobody asked.... I prefer my spare oars with blade towards the stern and handles forward. The handles make a nice blade rest. It's easier to get on and off the boat. If I need to deploy one, I just undo ONE cam and slide the oar out from the front. It takes a moment longer to shove it into the oarlock, but it takes three moments FEWER to release it from storage; overall, it works.