The names of best Universities or Institutes for Ph D Studies in India as your cousin wants to take admission in it are as follows:

Indian Institute of Science (IISc)
Indian Institute of Technology Bombay
Indian Institute of Technology Delhi
Indian Institute of Technology Madras
Indian Institute of Technology, Kanpur
Indian Institute of Technology Kharagpur
Tata Institute of Fundamental Reseach (TIFR)
Chennai Mathematical Institute (CMI)
Indian Statistical Institute Kolkata (ISI)
Hyderabad Central University (HCU)
Jawaharlal Nehru University (JNU)
University of Roorkee
Birla Institute of Technology & Science
Anna University, Chennai
University of Mumbai
University of Delhi
PEC University of Technology, Chandigarh
Amity University, Noida
Global Center For Clinical Research, Hyderabad
Manav Rachna International University - Faculty of Engineering and Technology
Jaipur National University (JNU), Jaipur
Christ University Faculty of Engineering, Mysore Road, Bangalore