What can be done with left over mussel shells?
The leftover shells of many different shellfish are excellent to use as cooking vessels. They clean up nicely and can be reused over and over if cared for properly. Bonus, they make a great presentation.

I use scallop shells for stuffed scallops (entree portion) and other casserole-type entrees, stuffings, and some side dishes. I use medium sized clam shells for stuffed clams, stuffed scallops (appetizer portion), clams casino, etc. They can be used for both appetizers or entree portions. Small clam shells work well for appetizers.

Mussel shells are perfect for - of course - stuffed mussels. They also make a great cooking and serving vessel for many other appetizers. (I use them mostly for appetizers because of their size.) 

You can be very creative and the actual mussels, clams, etc. don't have to be the main focus. E.g., I may have a fish fry and use shells for appetizers or sides. The smaller shells, like mussels, also make a quite nice vessel for individual condiments. Kind of a theme and quite unique. 

Note: Most everything I use the shells for goes into the oven or on the grill. Therefore I put a very light coat of oil or butter on the shells so the food doesn't stick. For anything cold, the oil wouldn't be necessary.

You do want to wash and dry the shells before the first use and then after each subsequent use. Larger, heavier shells I wash on the top rack of my dishwasher. Smaller or thinner shells I hand wash and dry.

Some people do boil their leftover mussel shells with herbs and seasonings of their choice to make a seafood stock. And, of course, it can be reduced to use as a sauce or as a part of one.

If you intend to make a stock or sauce, I would refrigerate the shells immediately after your meal and keep them refrigerated until you use them. In this case you wouldn't want to wash them first. But if you intend to reuse them, be sure to wash and dry them thoroughly.