Messages won't let me sign in to iMessage
How to Fix Messages
The following steps worked for me:
Update December 2016 - this happened once more - this fixed it again, so I'm even more certain of it now.

Clear PRAM (NVRAM)
Hold Cmd ⌘+Opt ⌥+P+R at the chimes until they are heard for a second time, then release

Start in Safe Mode
As soon as you hear the second chimes hold  Shift ⇧  &amp; keep holding until you see the regular startup progress thermometer

Attempt to login to iMessage.
Mine at this point appeared to fail &amp; the iMessage login screen just went grey.
I stared at it for 5 minutes, then rebooted.

Done.


Recommended but Unsuccessful
These things were recommended elsewhere but they did not work for me:

Checking Firewall settings til I was blue in the face.
It worked yesterday &amp; still works for all other machines in the house...
Changing DNS provider.
Logging out of everything iCloud related then back in again.
Any amount of messing with the keychain - Repairing, Erasing[1]
Restoring random items in ~/Library/Preferences; including, at one point, the entire folder
'Repair install' using latest Combo Update.
Recovering entire drive from a day-old known-good clone.
Reinstalling entire OS from Time Machine.

I'm copying this answer to all relevant Stack Exchange questions, in case it helps others. You would not believe the amount of chasing round I did to fix this one ;-)

[1] Many articles (&amp; Keychain Repair itself) claim that com.apple.security.plist ought to be owned by the user - in fact every time you fix that, next reboot it reverts to being owned by root.
I gave up trying to 'cure' that as it appeared to make no difference in the end.