Vehicle makes a rattling sound underneath (mid/rear end area?) when hit bumps and mechanic said may be the anti-sway bar (or front struts). What is that and how important is it to fix? Vehicle has 108,000 miles.
The noise might caused from a broken sway bar end link. This is what connects the sway bar to the control arm. If the link is broken, each time you hit a bump the control arm rises up and hits the broken link causing the rattling noise.
There are a lot of things that can make a rattling sound when you go over bumps. If you are serious about locating it, then your mechanic will need to test drive the car and maybe perform some tests, for which he is justified in charging a fee for.
However, based on your description of the location, you should open your trunk and open the spare tire bay. Something could be rattling around in there, the spare might not be secures or the jack and lug wrench might not be secured. If all seems secure, you might try removing the spare just to see that there is nothing under it.
Also look for things that might be under the seats. I just realized that your spare tire is hung under your car so my first paragraph does not apply in your case, but it would apply to any cars you own. The chain that your spare tire is hung from may be the source of your noise, or the spare might even be a little loose.
You might be missing some of the doughnuts that your exhaust hangs from allowing it to move around. This is something your mechanic would look for, I hope. But it could also be a sway bar end link or a strut mount.