An explanation of this is that they wanted to not forget the exile Jews have a thing for not letting go of past tragedies, and much of Jewish tradition is to do with remembering these events, such as the Egyptian period of slavery and the exodus, and the Babylonian exile.
Note that the same lack of patern can be seen in the name of weekdays: So for example, most of them are from the Babylonian calendar, via Arabic, so why didn't they use ayar for May?
What's up with the Turkish months? Surely it is just as surprising that Hebrew took it in as is, defying monotheism in a way, as it is that Turkish took some names here and some there.
Tammuz is certainly of Babylonian origin, as are the others.
It'd all make a very nice read if someone were to check all this and put it together in an article of sorts. Nothing of this is a trustworthy source, i don't think - but i doubt there is deliberate misinformation in it.
Other than this, it could be that the Jews simply liked or found it convenient that the months had names and continued to use them. That second link i put indicates for example that Tammuz is named after a Sumerian and Babylonian god of that name. Unless you have another etymology proposition for Tammuz in Hebrew?
We borrowed some of the month names from Greek: And here's another page about months: I don't really know why we didn't use all Arabic or all Greek. It all sounds pretty fantastic really, but what we see as a crazy patchwork is probably felt by native speakers as obscure and quite irrelevant etymology for everyday words.
And you're totally correct that our own system is strange.
It's not so surprising that Hebrew has the Babylonian calendar system, considering the whole Babylonian exile thing around 600-520 BCE. As far as guesses go, i don't think it's overly surprising to see similarities between Hebrew and Arabic, especially when it's acknowledge that their origins are in Sumerian or Babylonian.
From the Gregorian calendar: Yes, I noticed this too. I guess it goes to show that these names are not the product of careful forging by ancient wise men, but rather that of usage and tradition, colliding sometimes. I just quoted that link i gave on top, and i think the author there reused someone else's stuff. Before this, only four months were mentioned in the bible: