Evgeni Leshtanski's answer clued me in to what was really happening. The question isn't, "How do I tell some random prompt what my password is?" but rather, "What is going on?"

In my case -- and this probably affects a lot of people -- the underlying culprit was Empathy. I don't use Empathy, so I don't want it to authenticate to Google Talk on my behalf. Now here's where it gets interesting. When I followed Evgeni's lead and ran Online Accounts (under System Settings), it was completely unpopulated. That's because it auto-populates when apps that hook in to Online Accounts update credentials. Since I never filled in that Access Prompt dialog box, this never happened and I had nothing to change.

The key action that's leading this behavior on startup is that Empathy defaults to "Automatically connect on startup" even if you don't actually use it. So run Empathy, go to Preferences, and under the General tab, uncheck "Automatically connect on startup".

In my particular case, I had to also stop Evolution from checking my Gmail account before this dialog box went away completely.