This reminded me, I was going to ask the same question a while back.
It can be quite embarrassing when you've had some kiddie "trash-talking" you and the message sits there on the HUD, for your family to see.
Also doesn't help that it might have happened several hours ago and has long been replaced with NPC messages but then, the next time you log in, the game seems to re-post the last message you had from another CMDR.
Nice to know they can be cleared. [up]