Homer's two epic poems follow the story of the Trojan War through various perspectives. Did Homer make up the stories, or was there some kind of historical/mythological predecessor that he retold (or exaggerated, if the answer is historical)?
I'm going to start with a minor framing correction, before I go on to actually answer your question.
The Iliad and The Odyssey aren't texts written by Homer. They're actually oral traditions. It's even debatable whether anyone named "Homer" even existed at all, but if they did, certainly all they did was transcribe a purely verbal story that was passed down for generations.
So, the major correction to your question is: The Iliad and The Odyssey are a part of Greek mythology. They didn't arise from Greek mythology (any more than mythology arises from itself), because in oral tradition mythology, story and history quickly blend together. Stories like these may have traced real events, but they were educational tools about good manners, how to treat guests, marks of cultural significance, and just generally how the gods affect peoples' lives. These stories get told and re-told; they accrue variations, adopt easily memorizable rhythms, and underwent significant changes as time went on.
So, I'm going to go out on a limb and rephrase your question for you: did The Iliad and The Odyssey develop around historical events? This question is actually more directly answerable. Note, though, that it implies that you have to take any historical basis with a grain of salt. Remember that these are oral traditions; just because they once told stories that happened doesn't necessarily mean they still do.
In the case of the Iliad, the story as written is removed from its transcription by roughly four centuries. However, it does appear to have alluded to some real events and places, which means that it might actually have a historical basis. The city of Troy did exist, and the geography described matches the region. It likely corresponds to modern-day Hisarlik. "The Trojans and Their Neighbors" delves into this topic further, drawing parallels between story elements that appeared in the Iliad and descriptions of peoples and cultures that appeared in the area. Additionally, it seems that helmets described in the Iliad were in use at the time. However, there isn't enough evidence to draw the conclusion that the Trojan War happened as described in our transcription of events. There is barely enough to conclude that a real people were described.
The Odyssey is a little more up in the air. Most of the Odyssey is spent describing events which could not possibly have happened in our non-mythic consciousness understanding of the world. However, we can plausibly map where an Odysseus might have gone. Additionally, some archaeologists believe we have found Odysseus' palace on Ithaca. There's also a lot more evidence littered around in various places in the Odyssey - for example, it's believed that a solar eclipse is described in the Odyssey, which could possibly be used to date the events.
So, did the journey happen as written? Definitely not. But was there a person named Odysseus who left the Trojan War and returned to Ithaca? Honestly, it's quite possible. But one of the sources I found puts it best:
Why should we think that anything Homer wrote was historically accurate? It is mainly because he wrote about things of which he could not have had first-hand knowledge, but of which we have archaeological knowledge. Why should we think that the story is not totally accurate? Because we know at least of anachronisms, of things described that did exist as he described, but not at the time of which he writes.
This is a broad topic, and this post only scratches the surface of it. Hopefully I've given you both reason to consider that certain elements may be true, but don't take the historicity of the story with anything more than a grain of salt.
There are many opinions who Homer was, man or woman, bard etc. I think Homer was Odysseus because his descriptions were too perfect, like he was witness to those events. The Iliad started with Odysseus and continued with him in the Odyssey.
Odysseus's island Ithaca (better say Penelopa's island) is today's island named Unije off the Croatian coast. It is posted on the WEST of the archipelago Kvarner (really Kephalenia), following Homer's descriptions. This island is geographically and linguistically adequate for the claim that it is the Homeric Ithaca. One interesting thing: near the coast there is a village named Homer (Gorski kotar, Croatia). Nobody knows why.
Yes, Homer's epics talk about historical events.