Are you sure you don't mean the U.S. Mint? The Franklin Mint makes and sells "collectibles". They used to have a museum outside of Philadelphia, but that closed in 2004.
I second this! If you are going to DC during the week you are normally better off staying near a Metro outside the city. This may not be true for Christmas week though.
One place you might consider staying is Old Town Alexandria (in VA). It is a quaint colonial city with much of the architecture still in place. There are quite a few shops and restaurants in Old Town as well. I'm not sure it would be cheaper, but there would be more to do at night.
Another area that would be nice to stay is National Harbor in MD. It is just across the river from Alexandria. It does not have the Metro within walking distance, but they do have a shuttle to a Metro station. Again, may not be the cheapest area, but more to do. For example, the Gaylord National has an ice sculpture village around Christmas time.
I'm not sure what CTdiznymom is referring to by the attractions not being open on the weekend. As far as I know, everything is open everyday, year round. Maybe not the Mint, but definitely the museums and monuments would be open on the weekend. (You did mention coming on Monday though!)
I would also add a trip to the National Christmas Tree one night. It is on the ellipse in front of the White House. They have the National Christmas Tree plus 50+ little Christmas trees around it (one for each state and territory). They also have a train layout, a menorah, and a large yule log. They have choir singers too on certain nights.