I don't think anyone can say for sure when the myth of Prometheus first came about. But I can help shed a little light on the significance of his liver being eaten. It doesn't have much to do with a liver's regenerative properties and has more to do with the cultural value of the liver in Ancient Greece. The liver was considered to be the seat of the soul. Similar to how later cultures claimed the soul lived in the heart or the pineal gland. The eagle devouring Prometheus' liver was actually eating the physical manifestation of his soul every day. That was why it was considered truly torturous.
No. The Prometheus myth goes back at least to the early 7th century BCE, in the Hesiodic Theogony and Works and Days, which is centuries earlier than any evidence for the (extremely rare) practice of human dissection for research purposes. The bit about the liver regenerating is at Theogony 524-5.
Incidentally, the Hesiodic versions are also more than 200 years earlier than any evidence of the liver being thought of as the seat of the passions, or of intelligence. That interpretation isn't actually impossible, but there's no real support for it.
M. L. West, Hesiod: Theogony (Oxford, 1966), pp. 313-15, argues that the myth is derived from a Caucasian myth that has an evil giant bound to a pillar or in a cave suffering eternal punishment (the fate of Loki is another example of this story-type), but he doesn't have any good suggestions about why the Hesiodic account focuses on Prometheus' liver.
It turns out Prometheus isn't actually unique, though! Devouring someone's liver as a grotesque punishment is a trope that recurs in a couple of other early Greek texts. So we need to think about Prometheus' punishment in that context.
Tityos suffers the same punishment in the Underworld according to another 7th-century source, Odyssey 11.578-9). We're not told that his liver regenerates.
Also in a 7th-century-BCE source, Hekabe declares that she wants to devour her enemy's liver (Iliad 24.212-13), and in context it's clear that this isn't because she wants to inflict permanent punishment on him, but rather because she wants to desecrate his body after he's dead.
Similarly, the same enemy (Achilleus) had previously wished that he could devour the flesh (not specifically the liver) of his enemy (Iliad 22.346-8). The desecration of enemies' corpses is a trope that escalates throughout that poem (see C. Segal, The Theme of the Mutilation of the Corpse in the Iliad (Leiden, 1971)); it looks very much as though this devouring of an enemy's flesh is that trope taken to an extreme.
The Iliad attestations strongly suggest that eating the liver is not so much an attempt to devise a physically plausible scenario for eternal repetition of the punishment, as a standard trope for desecrating the body of a particularly hated enemy. The regeneration and repetition are most likely simply a miraculous way of making it more gruesome yet.
Edit. Perseus messed up on understanding the link for the Odyssey passage; now solved.
Unlikely. Regeneration in Greek, and other ancient mythologies, isn't really anything remarkable to the Prometheus myth. Because deities were immortal, and by extension their body parts, there are a lot of myths that include examples of regeneration. In fact, I'd be hard pressed to think of a mythology that did not include at least one regeneration myth.
Prometheus is the most prominent because a) a single body part is regenerated over and over and over again rather than an entity being restored once from an extant body part, making it a particularly gruesome story b) the subsequent discovery that the liver can be regenerated, and c) the numerous Promethean references with respect to stem cell research these days.
The earliest references to Prometheus in the written record appear in Hesiod's Theogony in the 8th century BCE. This would have been based upon even earlier oral tradition placing the myth before the era of "natural philosophy." Hippocrates was a good four centuries or so after Hesiod as well.
This is a good article that addresses Prometheus' linkage to liver regeneration and whether or not the Greeks knew about it. Professors Rasko and Power find it unlikely that they did and provide some decent arguments for their assertion.
http://sydney.edu.au/medicine/news/pubs/radiuscontents/2009/September/22_3_prometheus.pdf
Is it really true that the Greeks discovered liver can regenerate? How would they have discovered that? They didn't have surgeries back then.