Does IITM offer minor programs and how useful is it?

Yes, IITM does offer minor programs. Offering academic flexibility is something the university is great at. In short, students can choose almost 50% of their courses. If a student chooses 4 elective courses in one contiguous area, they will get a minor in that field. For example, if a student chooses four courses from Economics, s/he will be conferred a Minor in Economics. There are several such courses, ranging from sociology, philosophy and economics to Artificial Intelligence and Machine Learning. More details can be seen here. In addition, there is the option of studying an entire semester in a foreign university.

These are very useful in placements and in making career plans as there is added flexibility when choosing higher studies. Sometimes, a combination (such as mechanical + economics) is helpful, helping students become even more specialized in certain roles. Of late, we've seen interesting major-minor combinations be sought after by recruiters.