I am running Debian Linux that connects to my router
via DHCP client. The router picks up the DNS servers (or acts as a DNS relay) with my ISP nameservers. Unfortunately, my ISP DNS servers fail to resolve a lot of names they should. How do I add 1 or 2 lower priority nameservers (lower priority than my ones obtained from DHCP) as backups to my machine. I would enter my ISP's DNS servers manually into my router, but they do seem to change sometimes when my connection resets. I am somewhat a noobie, so I need direct instructions. Also, any good suggestions for reliable, public nameservers in the northeast, US?

BTW, I found the thread:
How do I assign an additional nameserver while using dhcp?
but there was no real resolution answer to this problem.

In your current scenario you can't, unless you can tell your router to not supply DNS servers (or to supply alternate DNS servers other than what is passed to it when it connects to your ISP).

One way to do this would be to make your debian box the DHCP server. Then you can select any DNS servers you'd like or use the ones that your ISP provides. Of course, this won't work if you are dual-booting this box.