My friend had options of Ph.D. from JU, IIEST as well as IIT Patna and still chose IIT Patna. She claimed that IIT Patna had a better research culture and the professors get work published in very good journals, A/A* conferences, etc. The professors motivate the students for the same and this aspect attracted him the most. However, your personal preferences also come to play here. So consider the following points
Choose IIT Patna for its research culture and not the IIT tag.