There's a lot of speculation on Finn's origins, which I have collected here for your perusal. Why all the speculation? Reasons I have heard are:
Whatever the cause, judge for yourself.
Theories
Finn is Prometheus
Greek titan and deliverer of fire.
Prometheus (forethought) was the titan who helped create the creatures of earth with his brother, Epimethius (afterthought). Epimethius screwed up while doling out gifts to the races, leaving the infant mankind without any form of defence. Prometheus, nice guy that he is, makes up for it by stealing fire from the gods and giving it to mankind. While Epimethius got the first woman, Pandora (who inadvertantly condemned mankind to a life of misery), Prometheus was chained to a rock by Hephaestus and had his liver eaten out by an eagle every day until he told Zeus which woman not to sleep with if he didn't want to be deposed from his own throne. Once Prometheus answered Zeus's question, Zeus allowed his son Hercules to release him from his imprisonment. In deference to the fact that I'm using Chiron, I also count the myth that says Prometheus accepted immortality in Chiron's stead when a poisoned arrow was dropped on the centaur's foot (by the self-same Hercules who freed Prometheus, surprise surprise), saving him from the alternative of eternal suffering.
Finn is Fionn MacCumhail/Finn McCool
Poet, magician, Irish folk hero.
A man named Finegas had spent much of his time trying to catch the Magic Salmon of Knowledge-the source of great power and wisdom to whomever should catch and eat it. After a great struggle, Finegas managed to catch the salmon. He told Fionn to light a fire and cook it. Finegas told Fionn not to taste even a mouthful of it. To see if the fish was cooked, Fionn prodded the fish with his finger and was burned. Without thinking he put his finger into his mouth to ease the pain, thereby tasting the fish and gaining all knowledge. With the blessing of Finegas, Fionn MacCumhail went on to become a great leader of Ireland. Aided by his companion Bran, Fionn chewed his knowledge-imbued thumb to get himself out of tight spots, such as giant-attacks, hags, arsonists, and magical decapitated catheads.
(Many thanks to the M for help in the summary. Irish folktales have a unique brain-melting quality.)
Finn is Oedipus
King of Thebes, root of an irritating Freudian theory.
Oedipus was the son of Laius and Jocasta, who, having been warned by an oracle that any male offspring would kill his father and marry his mother, turned him over to a herdsman to expose on a hillside. Oedipus was found by a horseman of Corinth, who was brought to and healed by the queen. Later, when Oedipus grew to manhood, he was warned by the oracle at Delphi that if he returned to his homeland he would -- surprise surprise -- kill his father and marry his mother. As he had not been informed that he was adopted, Oedipus resolved never to return to Corinth. Unfortunately, he crossed paths with Laius, himself on the way to Delphi. After a skirmish involving who had the right of way on the narrow rode, Oedipus killed his father. The next king of Thebes, Creon (Jocasta's brother), eventually ran into some trouble with a sphinx who had taken to slaughtering people and ruining the land. Evidentally he became rather desperate, for he decreed that anyone who could get rid of the sphinx would inherit the kingdom as well as his sister's hand in marriage. Naturally, Oedipus did so when he answered the sphinx's riddle ("What is that which has one voice and yet becomes four-footed and two-footed and three-footed?" "Man."), causing the monster to hurl herself from a cliff. One thing led to another, and it was not before he fathered four children on his mother and a plague had befallen the land that Oedipus realized his error. It so happened that Oedipus' adopted father died, which prompted the queen to finally inform him of his adoption. Putting two and two together, the herdsman who had exposed Oedipus as a child came forward and named him the son of Laius. The result was less than contained; Oedipus immediatly put his own eyes out with his mother's brooch, and Jocasta herself committed suicide. He eventually split the kingdom between his two sons and left for Colonus at Attica with his daughter, Antigone, where he died. Oh yeah, and there was some stuff about the ensuing wars and such, but if you really want to know about that I suggest reading Sophocles' trilogy Oedipus Rex, Oedipus at Colonus, and Antigone. (Hey, I had to.)
Finn is Herakles (Hercules)
Demigod, hero, kind of a jerk.
Herakles is the mythical son of Zeus and Alcmena. You know, the guy who Hera drove insane and caused him to accidentally kill his wife and/or children, for which his famous twelve labors were atonement. His exploits are too many to name, but he is notable for making enemies of both Hera and Apollo and having killed his way through half of Greece. He "died" when his wife Deianira gave him a tunic soaked in the poison blood of a centaur her husband had killed to save her, but attained god-hood by lying on a lighted pyre that burned away his mortality.
The following summary of supporting reasons was taken from an email by MGT:
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Both are capable of prodigious physical feats far in excess of what
they would appear to be able to do.
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Neither are particularly noted for outstanding intelligence, but are
(sometimes grudgingly) noted for extreme heroism, fearlessness and dogged
determination once a 'task' has been set upon.
- Heracles in the original myth was not seen as a muscle-bound "hulk" but
rather as a more normal physique, not unlike Finn. His power came from
his divine origins as a son of Zeus fathered on a mortal woman. The
"Hercules as Hulk" version was a later Roman corruption of the myth which
has been perpetuated into modern times.
- Heracles, being a son of Zeus, naturally became a target for Apollo's
jealousy. This may help explain Apollo's instant and intense dislike and
distrust of Finn over and above his customary disdain for mortals.
- Heracles fought directly with Apollo over the Delphic Tripod - could
this relate to Finn's freeing the Sybil from her deadly vision-loop?
(Prophecy and Oracles being the domain of Apollo.)
- Heracles fought indirectly with Apollo through Apollo's various
surrogates many times. One example being Heracles' terrible (but
victorious) battle with Cycnus (according to Hesiod).
Got a theory? Feel free to submit it.