An automated "backup" service for your website that crawls around and builds a static copy of all the pages it can see. If the site goes does (or if you push a maintenance mode button) it switches to the static site, maybe with a small warning message. You'd probably need to run the DNS or integrate with DNS services to make that easy to implement. The key is it needs to be stupidly easy to set up. (ping me if you ac…
Very similar, but not exactly: Cloudflare. Takes over your dns routes and will detect when you are down and tries to continue serving.
Also there's a lot more you could do. For example, versioning: show me what the site looked like last thursday. For people using crappy or nonexistent CMSes this could be huge.