Adding a short string to your packets isn't going to be a big deal unless you're sending them at a very high rate. And for a simple game, you're probably Doing It Wrong if you're sending that many packets. The only thing that you need to be concerned about is exceeding the MTU of your network, which you probably won't because it's about 1500 bytes for most networks these days.
If you really think that you can't afford a few bytes for a peer name, you could try using the remote socket address (IP address and UDP port). Don't use only the remote IP address, because it may not be unique. I'm not sure off hand how to get that in C#.