I'm writing a book that takes place in modern day but also involves the Greek gods and goddesses and Greek mythology. Is it frowned upon to change a small part of a certain myth to make it fit into your story? For example, I want to incorporate a certain minor god from Greek mythology in my book but, in the myth, he was killed as a child. Can I write that he somehow came back to life and lived to adulthood in secret, so he can help my main character in my book?
Not even the writers of the classical period could agree on a common canon truth. There are plenty of stories which contradict each other. For example, Aphrodite was the daughter of Zeus according to Homer but the daughter of Uranus according to Hesiod.
If you would like to add to this ancient shared fictional universe, enjoy the same creative freedoms writers had for millennia. If you can't find some detail in the sources available to you, just make something up. If what you found doesn't fit into your story, just change it. May Apollo forgive you (unless art and poetry are the domains of some other god in your story).
It's your story. Myths are stories. Change them however you want to.
For instance, there have been several stories written using the Norse mythology. Each of them treated everything differently.
There was "Day of the Giants" by Lester del Ray. It's an excellent read, with two modern (at the time of writing) twins taken into Asgard. One was a mercenary, chosen by Thor, and one was a rather bookish sort, chosen by Loki.
Sleipnir was a book set in a more modern setting (only 20 years ago or so) and the hero has a hated of Odin because he seems to be taking people he isn't entitled to, such as his best friend. So our hero goes into a deep cave where he hopes to find Sleipnir to get to Odin.
There are many others. Each of them makes changes to the mythology in their own ways.
Roger Zelazny wrote a short story, "The Last Defender of Camalot" which changes the Arthurian mythos. I recall another short story in that mythos set in a somewhat cyberpunk future. There was a comic series (Camelot 2000?) which had the main characters of the mythos reincarnate to save a future Brittan from alien invaders. Except for Arthur, who had been preserved, the rest were modern people who had suddenly remembered their past lives. The comics "Mage," "Mage 2," and "Mage 3" use this mythos as it's base.
I'm currently writing a story that makes strong changes to the Arthurian mythos and the Celtic mythos.
Absolutely. Not only do the canonical texts disagree as Philipp stated, but the ancient Greeks themselves typically based theatrical performances on their myths, sometimes with parodic intent. So doing variant versions of the stories have existed as long as the stories themselves have.
I would urge you (and it appears you are doing this) to educate yourself on the myth you are reworking. I passed on a collaboration once because a co-writer's plot involved claiming two entities were the same individual; I had no moral objection to doing this but I felt his manner of doing so misunderstood what the entities represented, and the story would have been the lesser for it. These stories have lasted so long because they are complex and resonate deeply with us and usually deeper research provides hidden ramifications which will make your story all the better, and your alterations will benefit as a result.