Wikipedia notes that many ancients believed that there were hollowed-out areas underground, where places such as Hell (or the underworld in general, hence the underworld) existed. But these weren't really "Hollow Earth" theories. We didn't get that far until a few hundred years ago.
This attributes the modern Hollow-Earth hypothesis to Edmund Halley, the famed astronomer, circa 1692. He was a brilliant guy, but this was not exactly his most brilliant idea. He proposed the idea after noticing some strange properties of the Earth's magnetic field:
Halley was fascinated by the earth's magnetic field. He noticed the direction of the field varied slightly over time and the only way he could account for this was there existed not one, but several, magnetic fields. Halley came to believe that the Earth was hollow and within it was a second sphere with another field. In fact, to account for all the variations in the field, Halley finally proposed that the Earth was composed of some four spheres, each nestled inside another.
This isn't too bad, but the idea of proposing loads of different spheres to account for corrections in observations reeks of epicycles. But it gets worse:
Halley also suggested that the interior of the Earth was populated with life and lit by a luminous atmosphere. He thought the aurora borealis, or northern lights, was caused by the escape of this gas through a thin crust at the poles.
The site goes on to discuss those who came later to develop the theory: John Symmes, Sir John Leslie, and even (gasp - don't look unless you want to be really disappointed)
Some of the more recent proponents didn't have such good reputations. Take Cyrus Read Teed:
Teed changed his name to Koresh and founded what might today be called a cult. After buying a 300 acre tract in Florida, Koresh declared himself the messiah of a new religion. He died in 1908 without proving his ideas.
A second source gives a similar discussion on the subject. It attributes the modern-day theory to John Symmes (mentioned above):
The hollow earth theory actually seems to have been originated in the early 1800s by John Symmes, an earnest American who devoted the greater part of his later life to convincing the world that the earth was formed by a series of concentric shells.
This page gives an interesting discussion of the idea. At first, I wasn't sure if it supported the theory or not, but it appears to go against it. It, too, says that Halley was the first to develop the idea, but it does use the two pages I listed above as sources, so this could be a duplicate.
John Symmes appears to have been one of the proponents of the modern theory, but we can trace it back all the way to Edmund Halley.