Thank you for your reply, Matir!
The router does accept pings from other clients.
This morning, I went into Debian and switched the network configuration from static to DHCP and the internet immediately began to work. It can also see my router now.
So, given my low level of network understanding, it seems like my router might be blocking the internal IP of the one computer (since I've got it set for DHCP addresses to begin as slightly higher numbers). I've got a fairly common D-Link DI-604 and just went through each page and option in detail - everything seems to be in order. Does anybody have any ideas of specific places to look?
UPDATE: I tried pinging from my router (via the web interface) to my problem computer and it didn't work. Thinking that that was really strange, I rebooted my router and everything seemed to work! Moral of the story: when in doubt, reboot the router.