To what extent does college name matter. Should I drop this year, or join IIIT Delhi CS?

Dropping a year would not be a wise decision if you are genuinely interested in CSE, and have an option of joining IIIT Delhi. I don’t think the college name matters to the extent that you would be giving up on the chance of joining IIIT Delhi CS. In fact, it’s slowly making its own name and in a few years will be among the top engineering colleges in India. Moreover, your UG college is just a stepping stone and its brand name won’t matter much and does not correlate with one’s success.

Joining IIIT Delhi will open doors to multiple opportunities like working on big projects, exploring hobbies, and traveling. If you choose to take a drop year, you will be struggling again with the same exam just to get a better rank. Personally, I don’t think these exams are authentic measures of an individual’s aptitude for computer science. 

A close friend of mine was in a similar dilemma and was about to take a drop year when his mother persuaded him against the decision he finally decided to join IIIT Delhi and later realized it to be the best decision of his life. The 4 years were an ethereal experience allowing him to explore his research interests, learn and apply what’s new in computer science, and grow as an individual. So, in your case too, taking a drop year isn’t advisable.