@TomWolsky: Thanks for following up. I used to have that same issue more frequently as well. It's less common now, but I still from time to time am prompted to login to iCloud randomly — at least from my perspective.

Fix for Mac Randomly Asking for Password to iCloud, FaceTime, Messages. A notable amount of Mac users have discovered that a random OS X password popup window can appear from iCloud, FaceTime, or Messages, any of each asking for the respective iCloud password. The random password request is very non-specific and just has the logo.

Apple Mac Keeps Asking For Icloud Password

I'm trying to think back to if there was something that helped me in the past, but I suspect it may have been just on Apple's end. Interestingly, not long ago there was a similar issue on the Mac with iCloud tokens expiring and requiring a fresh login at least daily. That too was affecting 1Password users (myself included) who had setup iCloud syncing, but Apple was able to resolve that.

Iphone Keeps Asking For Password

Mac Keeps Asking For Icloud Keychain Password

This is pure speculation, but it seemed like maybe they were making some changes behind the scenes before WWDC, perhaps to prepare for the developer previews of macOS (née OS X) Sierra and iOS 10. Similarly, I can only imagine it's difficult to introduce and support new iCloud features and continue to support users on older OSes without a few glitches. I suspect that they'll be able to iron out these kinks as well, but it may be worthwhile to contact Apple, if nothing else than so they're aware that their customers are having these issues.