Here are some things to check out:
1) iTunes must be open for Home Sharing to work
2) Verify that the ATV and Mac Mini are both connected to the network, and that they have compatible IP Addresses.
3) Make sure that Home Sharing is enabled
4) Make sure that the firewall on your mini is allowing iTunes to receive requests.
Since rebooting everything solves the problem, I think #1 & 2 are the only things that are really possible...