Why is the liver the only internal organ of the human body to regrow?
Maybe it is due to two factors:


The liver is one of the few solid non-tubular organs. If a tubular organ is damaged, all the layers that composes it must regenerate. This layers usually have different cell types, which is always nasty for regeneration since some of them may be formed by specialized tissue (for instance, myocytes are very difficult to regenerate. If the organ has a muscle layer, such as the esophagus, regeneration will be more difficult). Furthermore, damage in a tubular organ usually involves perforation, which will disrupt its function even if the injure isn't big.
Hepatocytes are pretty undifferentiated cells. They are basically enzymatic sacks with some vacuoles and mitochondrion, but they remain as your typical animal cell. They have a regular nucleus, they don't have any cytoskeletal adaptation and their vacuoles don't compose the most part of their cytoplasm (in a healthy liver). Cells with so little morphological specialization usually regenerate with no problem. Plus, the liver itself is constantly exposed to chemical damage due to toxins, drugs and other chemical compounds that may be present in the diet. This situation is special and don't apply for any other organs, with the exception of the kidneys (which, in fact is a very complicated tubular organ).