If I have gotten anything wrong please let me know and I will edit accordingly
I'm not sure if there is a way to let this guy down gently, rejection is rejection even when it's sugar coated and there's no telling how he'll take it. He could be very gracious or he may not be, either way though, if your cousin isn't interested in him she should tell him and make sure he knows that before things get awkward.
If I were you cousin, I would find a moment either before or just after a class they have together to tell him that I am not interested in a relationship with him. She should be polite but to the point, and she doesn't have to apologize or justify herself. If she's not interested she's not interested, that's nobody's fault and hopefully this boy will accept that. I know this may seem harsh, but it's better to be honest and to the point as it helps avoid any confusion for everyone involved and means at the end of the day everyone is (hopefully) happy and the boy is aware that your cousin has made her decision.
She could say something along the lines of:
"Hey (boys name), I just wanted to let you know that I thought about what you said and I'm not interested, but I appreciated you being honest with me so I thought it's only fair for me to be honest with you too."
I see you mentioned in your question that they don't have any common friends so I'm guessing this means they don't really know each other either. If she wanted to, she could add in that she'd like to get to know him as a friend, it may help to soften the blow. I will add that she should only offer this to him if she is actually interested in potentially being friends with him, if she isn't then just leave it out, it's better for everyone.
I know that at the age of 16 everything can seem very daunting, but having been in the same position as your cousin a few times in my life I can say that being honest and direct is better in the long run even though it is a tough thing to do. It means that what you say is understood and everyone leaves knowing exactly where the other person stands, it leaves no room for something to be picked up the wrong way.
I am sure your cousin will be fine, good luck!