I've been dealing with the same issue after an upgrade from 10.6 to 10.10, although in my case there's the additional and possibly relevant hitch that my Google accounts are through a university's Google Apps service, so the standard Google login page always gets redirected to a University page to authenticate before sending you back to Google (regardless of whether you log directly into a Google service on the web or through an Apple app).
Oddly, I was still getting my email, and my calendar (both of which were configured prior to migrating) was also updating, despite the constant failed login errors.
I tried the suggestion of deleting and recreating the account, but that did not work. Re-creating the account failed on the same error.
I managed to get my email working again by adding it through Mail.app as a generic account, although the System pref pane recognized it as a Google account, and then started throwing the same errors.
After clearing caches, doing permissions fixes, and a few reboots, I spent some time staring at common log errors and noticed that both IdentityFinder (a university-supplied app that searches your drive for passwords, SSNs, and other identifying info you might have saved in cleartext) and Microsoft EndpointService (a virus scanner) were crashing and throwing oddball errors, respectively, in a steady stream.
I started with IdentityFinder and eventually managed to dig it out of LaunchServices, and after a reboot those errors stopped, the EndpointService errors stopped inexplicably as well (might have been an auto-update in there or something), and now I'm not getting the Google login errors. Whether it was removing IdentityFinder, fixing whatever the Ms virus scanner was doing weird, or something completely unrelated, something in there apparently fixed it. I'm reluctant to try deleting and recreating the account to confirm that it's 100% fixed, but it's strange that apparently unrelated things were interfering with this (maybe the virus scanner was interrupting the confirmation dialogue?).