Update 30 August 2020
It would appear this post is pulling a lot of traffic. As is typical in this type of documentation, the efficacy of this post has been overcome by events, specifically the full release of Debian Buster 64 bit for the Raspberry Pi 3 and 4. With the 64 bit version installed, Chromium doesn’t have the update dialog bug. Therefore, ignore my original post and instead update to 64 bit Buster. You can go here to find it: https://downloads.raspberrypi.org/raspios_arm64/images/
The parent folder has two folders, one dated 5-24 and a newer one dated 8-24. I started with the 5-24 release, which was and still is considered a pre-release. So far I’ve encountered no issues, and over time I’ve watched numerous updates come rolling in. I can only assume by now that I’m up to the 8-24 release. In any event, install the 8-24 release. Consider the 5-24 release only if all else fails.
I don’t know how it crept in, but there’s a bug in the current Chromium that keeps popping up a dialog complaining it can’t update Chromium. The only problem is that Chromium isn’t updated like regular Chrome, it can only be updated when there’s a package update via apt. And there are no updates. There’s a bug for this (see https://github.com/RPi-Distro/repo/issues/165), but if you’d like to shut up the annoyance now, then there is a work-around (I think).
In the file system under /etc/chromium-browser/customizations, open a new file named 01-disable-update-check:
cd /etc/chromium-browser/customizations sudo vi 01-disable-update-check
Then type in the following single line of text:
Save the file and restart Chromium. The interval value 31536000 is the number of seconds in 365 days, or one year. Why that number was chosen I can’t say. Nor can I say if the value really works, because restarting the browser always makes the nag dialog as well as the icon go away for a time on next startup. But we shall see…