I own an older row home (65 yrs old) in NE Philadelphia. The center area of my finished basement floor is sunken about 3.5" in relation to the perimeter of the basement walls. I always assumed there was a drain there underneath the carpet. It wasn't until I pulled up the carpet that I saw that wasn't the case.

The sunken center area wasn't a drain, but was in fact an area that the previous owners had used some self leveling cement to remedy.

I had a friend that does masonry work come and take a look at the floor. He drilled a couple of holes and we found out that there is a 6" gap between the poured floor and the dirt underneath. I guess from settling over the years. He says that is common to homes in NE Philadelphia. His suggestion was to cut out the floor, let it fall to the ground; and then pour a new floor over it. Does that sound like the best situation? I've read a bit about "mudjacking" and inserting concrete/cement through holes to re-float a floor. Is that an option for me? What should I be looking at?

Mudjacking is a method that has been used for many years to raise concrete. It is particularly good for slabs and if done properly can create a level slab, or in the case of an extreior application the drainage can be changed.

Each job is different, so it necessary to analyze the situation, particularly with an interior location.

Slabs can be leveled, but depending on the soil below, it may last for a couple of years or it could be relatively permanent. If this is an older structure and the settlement is completed, it can be a good investment. Often the price is less than people expect.

Mud jacking will only work if the floor is at least 4 inches thick and in fairly good shape. Also mud jacking can be expensive if done right and will only bring the floor up to fairly level position unless the guy is real good.
The primary reason for the gap between the floor can be multifold.
typcially when these gaps are created its caused by a few factors.
1. The contractor failed to remove any wood debris from under the floor before pouring it and over the years termites have eaten away the material causing the gaps.
2. The floor was placed over a clay based soil and as the soild dried out it shrank leaving the gap.
3. The builder dug to deep when digging out for the foundation and did not compact the soil properly before the pour.
4. There is underground moving water that undermined the slab.
5. any combo of above.
Your solution to the repair as concretemasonry said you need to analyze your situation but if it was me and we were doing the work.I would bust up the cement slab and remove it entirely. Some of the smaller pieces of cement can stay. Then new gravel, drainage pipes as needed. 6 mill plastic. Metal mesh and fresh coat of new cement at least a #4000 mix or better.
This is not a job for the faint of heart. Lots of heavy lifting and busting of cement.

We had aa similer issue with a home in southern NJ in a garage. We decided to mud jack it. It turned out to be a much larger hole then expected. We used over 5 yards of material to fill the gaps under the slab to push it up! Only a one car garage. We figured that the builder left a tree stump below the ground and covered it with cement rather then digging it out. After several years it rotted away and was eaten by termites and ants. We rodded the hole we drilled into the cement before we jacked and four foot of rod fell into the hole and disappeard.