One of my pilots has a permanent (?) low moral. It shows up as a red icon on the corner of his portrait.

I did not notice when he got it, or how. According to the tooltip, it causes his cost for morale-based commands to be higher.

Does anybody know how he got it? Or more importantly, how to cheer the guy up?

Yup, as vonHardenberg said, it's a bug.

Put the little p***y in a shite mech and send him out on a mission by himself to show him that there is more to be depressed about than not getting the last cup of coffee.

Well that blows! Think I'll keep him. I can't see this play-through lasting much longer since I've made so many ignorant errors in pilot advancement, anyway.

I've had a similar situation, two of my pilots have had the High Morale buff at "0 days left" for in-game months now. I have yet to play with the 1.0.2 patch though, it has just finished installing and I'm booting the game as I write this.

Someone on the Steam forums is stating that it is just a notification issue, in that the notification is not going away, but your moral is not being effected at all.

So if you were supposed to have high/low spirits for 30 days, on day 31 if that notification didn't go away and look at how much moral is being used for your moral skills, you will notice that it just uses up the normal amount of Moral

I can verify that this bug is fixed in the 1.0.3 beta version. It even removed a low morale status that was already stuck on a character in my save.