Murph and Professor Brand were working on a technology to manipulate gravity.
During the events of the film, they are trying to come to a theoretical understanding which reconciles gravity with quantum mechanics -- the so-called Theory of Everything (although it is not referred to by this name in the film). When
Cooper and Murph first visit the NASA facility
there is a remark to the effect that the entire facility is designed to function as a spacecraft, if it can be lifted into orbit -- which requires control of gravity.
Gravity-manipulation technology would ultimately allow the creation and control of wormholes, like the one seen in the film. More immediately, it could be used to lift large numbers of people off the Earth into orbital habitats
like the one seen near Saturn at the end of the film. Presumably, these habitats would be safe from contamination by the blight which was destroying all food crops on Earth.
In reality, a Theory of Everything would be necessary in order to manipulate gravity and control wormholes, but it might not be sufficient. Interstellar takes a few liberties here, which is reasonable enough as it's a science fiction film, not a physics paper.