The short answer is that Homer is clearly drawing on existing fables, stories, myths, and even epics to create his. This much is by necessity. The problem is that the details are and will likely always be unclear.
Alternate traditions abound
I think this is the clearest way to answer your question: not long after Homer, we have numerous stories that directly conflict with the Iliad. The most obvious differences are in names. In the larger Greek tradition, the mother of Oedipus was Jocasta, but in Homer her name is Epicasta.
One huge difference comes with the birth of Aphrodite. In Hesiod's Theogony, who was writing but a generation after Homer, the birth of Aphrodite was a primeval affair; she rose from the foam that was made when Uranus' testicles fell into the sea after Cronus cut them off. This is likely a nod to one of her titles: Aphrodite Urania.
However, in the Iliad she is clearly pictured as the daughter of Zeus, who mentions that her mother is Dione. There is no room for Uranus and a sea-foam birth here. Hesiod, if he knows Homer at all, gets this myth from elsewhere (if he didn't invent it).
Jonathan Burgess in his contribution to the Oxford Classical Dictionary includes another variation:
However, Herodotus (2.117) states that [in the Cypria] Paris and Helen sailed to Troy quickly, whereas Proclus indicates that they were blown off course by a storm to Sidon, which Paris sacked. For Herodotus, this is reason to doubt that Homer composed the Cypria, since Paris is said to have been at Sidon on his way home at Iliad 6.289–292. It seems that either Herodotus or Proclus is mistaken, unless the Cypria existed in variant forms.
Herodotus of course was reading the Cypria hundreds of years before Proclus and a couple hundred years before the Alexandrian scholars would begin the harmonization of the epic cycle.
Homer is an oral poet
This point is really important to keep in mind. Homer wasn't "writing" the Iliad and the Odyssey, but composing it orally. A few scholars suggest that Homer might have had access to writing, and maybe was literate, but even if that's the case, from research on oral tradition, it's clear that he composed this in an oral context first. This is why you get stock phrases like "rosy-fingered dawn" that peppers narrative transitions. You don't get these phrases in the Argonautica or Aeneid, whose authors were clearly literate and were clearly written down first.
Part of the oral epic poet, though, is innovation on the fly with the stock stories. Their methods were laid out in the research of Parry and Lord, who recorded Bosnian poets do the same.
It's too important to be made up whole cloth by one poet
The Trojan War is one of those fundamental stories that show up all over art and literature. It was so important that it colored their interactions with the Persians in the fifth century, festivals were made to recite it in the sixth century, and references outside Homer show up in the very next oldest poets in Greek literature (Hesiod, Hymn to Aphrodite, Arctinus, Archilochus, Sappho, Alcaeus, Stesichorus, etc.)
There really just isn't the time for it to have grown in importance so that the story would be known all over Greece if it were made up by a single person. Even if we assume he was a traveling bard, orally composing two giant epics from scratch without any starting point and then less than a generation later everyone writing as if the story is common knowledge without widespread adoption of writing is implausible and improbable.
So, yes, I'd say there's evidence that Homer didn't make it up.
However, which parts were taken from pre-existing traditions, which parts were adapted from other stories, which parts he invented, and which parts were tacked on later is all still up for debate.
[Parts of this are taken from an article I wrote on Homer and oral tradition that isn't online yet.]
