Just as @brhans mentioned in comment not all things that are not professional are unprofessional, some things just don't fall under this dichotomy.
Just like you've mentioned this person doesn't deal with clients and if dress-code is casual I can not see any reasons to try to deal with it in any way.
So I'd rather leave everything the way it is.
Unless, of course, someone for some reason is offended with the very fact that he/her is wearing black ribbon. Well, not necessarily offended, may it just makes somebody sad.
In that case I'd ask the mourning person whether he wants to work for a while remotely - not because someone is offended but because he is passing through hard times, just to regain emotional stability. Most probably he'll be back without a ribbon.
But again, the best option to just let it go. It will pass.