Jallaudin Akbar ruled some parts of India. During his empire he had Delhi (Modern day state) as his capital, but it was known by some other name.
What was the name that was used for the capital during the time that Akbar ruled?
http://en.wikipedia.org/wiki/History_of_Delhi#Early_history shows that the city was called Delhi (or Dilli) at least since the 12/13th century.
Therefore, when Akbar assumed throne in 1556, the city would have been called Delhi.
There are however more concrete sources than this simple extrapolation. Humayun Nama refers to the city as Delhi. You can read it here.
I am not exactly sure but I remember the media (in India) addressing Delhi as "HASTHINA". Is that what you are looking for ?
hmmm....i think "indraprasta" was the older name of delhi. But am not clear whether the name changed or not, during the period of akbar.