What is the value of an BU 1878 CC Morgan Silver Dollar?

The BU designation does not mean much. It is best to have coins professionally graded. That being said, the Carson City silver dollars are a great mint and that is a good year. A "brilliant uncirculated" could mean a variety of grades, but should qualify for at least a MS60 and go up from there. An MS60 can fetch $210.00 and an MS65 can go for $1810.00 so you see, it's best to get the coin "slabbed," or graded by a professional grading service. This will cost about $100.00 and can be done through your local coin shop.