Though it's certainly worth a shot, and I suppose Chrome could probably do a test query to 8.8.8.8 before recommending the switch.
I would prefer
"Your ISP (network administrator, ..) is intercepting and manipulating DNS requests. Do you want to install a local caching DNS server and use it instead?"
Unbound [1] is a local DNS server that I have installed on all my machines. Small, fast, security-oriented, IPv6, BSD, made by NLnet Labs.
If you use Google's DNS (or anybody's DNS) you are basically telling them which sites your are visiting. If you have a local DNS resolver you will not leak all that information to a single third party.
No, not a single upstream.
When my browser asks my server on 127.0.0.1 to lookup news.ycombinator.com it will first contact the root DNS server, then `com` DNS server, then then `ycombinator.com` DNS server. Who knows about the fact that I was looking for `news.ycombinator.com`? only the `ycombinator.com` DNS server. Who knows about the fact that I was looking for `ycombinator.com`? only the `com` DNS server. Now I go to slashdot.org. Who will know about that? Only the `org` DNS server. If you use the Google DNS, Google will know that you requested both `news.ycombinator.com` and `slashdot.org`. Do you want them or any single company to have all these information?
Obviously you need an upstream, an authoritative server somewhere. But why do you need to concentrate all these requests on a single DNS server? ISP are actively tracking users and, probably, selling their DNS histories to advertiser. Nominum (makers of a widespread DNS server) is quite explicit about it: «Data gathering and measurement are a vital part of network operations and DNS data represents a rich vein to be mined that has been underutilized in the past». [1]
"Your ISP (network administrator, ..) is intercepting and manipulating DNS requests. Do you want to use Google DNS instead?"
Comes in real handy when we need to have a "DnsManipulationDetector" in the future that checks if your DNS is actively censoring...