There are no written records, only oral histories, but the Illinois Parks Department does say that that is the origin of the name of the park:

http://www.museum.state.il.us/muslink/nat_amer/post/htmls/arch_starv.html

http://www.starvedrockstatepark.org/test/

The earliest place it appears in writing is in a Historical Journal in 1825: http://www.ao.uiuc.edu/courses/aiiopcmpss/StarvedRock/schoolcraft.htm

Yes and no. It's a complicated history, and thankfully Mark Walczynski untangled it a couple years ago in The Starved Rock Massacre of 1769: Fact or Fiction. In case the full article is unavailable to you, I'll summarize it.

The earliest record of "Starved Rock" being called such comes from 1834; before this it was usually called the Rock. At the time, there was a widely attested story describing the massacre of the Illinois at the Rock in 1769. As /u/JQuilty mentioned, this is the incident to which Schoolcraft referred in 1825. But there are a few other sources on it a well.

Judge John Dean Caton reported to the Chicago Historical Society in 1870 that he had heard of the incident from a Meachelle, "the oldest Potawatomi chief" present at an 1833 treaty signing in Chicago. In turn, Meachelle was supposedly a boy at the time of the incident. In the Caton/Meachelle version, the attack on the Illinois was in retaliation for the assassination of Pontiac. The Illinois fortified themselves atop the Rock while the Pontaic's old alliance besieged them. As hunger and thirst set in, the Illinois made a desperate descent down the Rock's sheer cliff, under the cover of a thunderstorm. After their descent, they were attacked by the besieging forces and almost the entirety of the Illinois population was killed. Only 11 survivors managed to escape to the safety of St Louis, and they eventually resettled among other peoples and faded out of history.

In 1873, local historian and politician Perry Armstrong related another version of the 1769 massacre. He attributes his information to Shick-Shack, who allegedly lead the besieging forces. In the Armstrong/Shick-Shack version, the siege was prompted, not by Pontiac's assassination, but a conflict over hunting rights west of the Wabash in which the Miami, Potawatomi, and Kickapoo allied against the Illinois. Four battles (on the Wabash, at Kankakee, at Blue Island, and at Morris) preceded the siege. Finally, the Illinois sought refuge on the Rock, but succumbed to smallpox, thirst, and starvation. In this version, only a few warriors made the descent down the cliff, but again most were killed in the ensuing battle - only seven escaped to St Louis. The women and children who remained behind were captured and divided among the victorious forces. Again, the Illinois cease to exist after this.

There are several problems with both accounts. First, no one else at the time seems to have noticed a significant conflict going on at the time - no traders who did regular business with the Illinois mentions this, no British, French, or Spanish officer notes significant forces moving through the region. Second, in 1769, the Illinois weren't living near the Rock but some 200+ miles away, so its a bit out of the way for them to retreat to when there were closer British and French forts to turn to (as they had in other situations). Third, the scattered bones of the fall - a frequent element in other folkloric accounts of the aftermath - have never turned up in many archaeological studies of the Rock. Finally, and most damningly if take these accounts literally, there's the fact that the Illinois weren't eliminated in 1769. Not only did they continue to be a common presence through the remainder of the 1700s, they're still around today (the Peoria Tribe of Indians of Oklahoma, for example).

However, all that said, there is a kernel of truth to the story. In 1721, during the Fox War, Minchilay - the nephew of the Fox chief Ouashala - led an attack against the French and Illinois. He and his men were captured and were executed by the Peoria, who were then living near the Rock. The following year, in retaliation, Ouashala organized another attack during which he besieged the Peoria while the fortified themselves atop the Rock. As in the other accounts, starvation and thirst set in, but rather than attempt to flee, the Peoria sued for peace. After a difficult negotiation, Ouashala and his forces relented and peace was resorted. This appears the true events (though some details of Ouashala's account might be exaggerated), and is not only attested through Ouashala's account, recorded slightly after the fact by the French, but also in concurrent French military orders as French soldiers were sent to relieve the siege at the Rock. After this siege, the Peoria abandoned their settlement near the Rock, which is why they wouldn't have been around in 1760s. As memory of the Fox War faded, the true events got tangled up in more recent history and folklore.