If yours has wi-fi (Pi 3 and above or Pi Zero W) you can always just leave it plugged in somewhere innocuous as a pi-hole. It's super useful and takes almost no time to set up, provided you can change the settings on your router to use it as DNS.
One option would be to set up two Pis, and distribute both IPs in DHCP. If one is down, or can't resolve, clients will fall back to the other one. The chance of both being down at once (unless it's something that takes your network down) seems low, and the cost of a Pi Zero W (~$10) wouldn't add much.
Even so, normally you can set a fallback DNS server on your router, and you could have two little Pi Zeros on a plugstrip in a closet somewhere! (Or just set the fallback at opendns or something)