The following was my response. I had never thought about the reasons behind continuing the study of Latin instead of a modern language in high school, but in thinking through it, I gained a fuller understanding of what it is I'm trying to do in my classroom.
It seems like a lot of the responses on Latin vs. modern language have been from lower schools only, so I thought I'd give some input from one who teaches both elementary and secondary Latin. Our students take Latin from 3rd -10th (we hope to offer Greek someday in the 11th and 12th). I find that Latin is especially beneficial to students transitioning from the Logic to Rhetoric phases. Elementary Latin is all about memorizing the vocabulary, endings, cases, etc. 6/7-8th grade Latin is about seeing the relationships between the words and learning how they work to form sentences the way they do. When students begin to transition into the more idiomatic/poetic Latin, it really helps them see how language works. This is around the time (and at our school even a little before) they study formal rhetoric. Ideally they should be reading some "real" Latin and becoming familiar with the rhetorical devices in Latin, how they translate to English, why they are effective, how the speeches impacted history, etc. These things don't make a whole lot of sense to students in the logic phase. They're still pretty concrete thinkers. In the poetic / rhetoric phase, it clicks with the students and they see the impact of language from a different perspective. Modern languages don't really offer these types of rich rhetorical histories which are so much a part of English's past. Modern languages are fine, don't misunderstand me, but if you're looking for the best language to use to train students in the tools of learning, I would say Latin is it. Your students are already familiar with it, and even if new students come into Latin it isn't so difficult it can't be made up with effort. I have a new 9th grader this year who has not had any Latin, while the other students have had at least 1 year, and one student has had 5 years. The new student picks up so much from the insight of the other students that he has done very well. He works very hard and currently has a B-. Most likely your students will have the opportunity (or even requirement!) to study a modern language in college. One of the best preparations for that is to teach them how to study a language (using Latin, which they are already familiar with!) using all three of the tools of learning.