Others have spoken of the concrete benefits -- Latin improves students' language skills, SAT scores, and even mathematical reasoning. More on that subject can be found here. My conviction is much more funamental: knowledge is good for its own sake. The mind is not a hard drive with limited storage space -- time spent engaging in intellectual exploration is never "wasted."
Studying Latin is an excellent way to build general knowledge. Foundational texts in almost every field of human endeavor are written in Latin. Education and job training often pigeonhole us into narrow specializations, but it is human nature to be curious about the whole world.
Latin is not the only "dead language" worth studying. There are equally robust traditions to be found in Sanskrit, ancient Chinese, Greek, Hebrew, and many other languages. If the Romans bore you, explore another tradition! There is more to be learned than can be learned in one lifetime.