Perones it could be a number of things causing it from post nasal drip, to an allergy or sinusitis.  It's also possible that it's to do with acid reflux.  It may or may not be something related to their diabetes. 

However, I understand how distracting this can be but I think it's something you're going to have to get used to.

My mother used to have a constant cough that used to annoy the hell out of me as it was like having a conversation with a cough not her.  For years I couldn't get used to watching television without a coughing sound in the background.  It turned out to be an indicator that she had cancer.  I'm not saying that the person you work with has anything as bad as this but just be mindful they may have a medical condition causing it.

Perhaps you could gently say something like aww you poor thing your voice/throat sounds really sore, have you had it checked rather than sitting there simmering.

And no you're not being mean just human   have a lozenge

Put those boots on

I want my boots back 

Kinkeh