And is there a better/non-political term to use than Palestine? But anyway, as I understand it Aramaic was the "native" tongue at the time of the Roman conquest(s) of the area. When and how did Aramaic die out and Arabic replace it?
For a bonus, what do we know of how/why Hebrew was replaced by Aramaic?
TIA.
is there a better/non-political term to use than Palestine?
Try "The Levant". That's the commonly accepted name for the area as a whole, rather than the transient political subdivisions which are of often contested usage.
As the other comment said, "the Levant" is a nice neutral term. "Palestine" at one point was a neutral geographic term, but now is often thought of politically. "The Holy Land" works one too.
The process of Arabic becoming the dominant language, and being Arab being the dominant identity, is not very creatively called Arabization. This was a gradual process, which occurred over the centuries following the Arab conquest of the region.
But, that's not the reason for my comment, which is for your additional question about Aramaic replacing Hebrew. Aramaic didn't just replace Hebrew--it replaced a number of languages in the region. In many areas, you can think of it as successive waves of language change--first, Aramaic gradually replacing a variety of languages in the Middle East, then Arabic doing the same to Aramaic.
The reason for this was the use of Aramaic as a prestigious language by a series of governments. Governmental languages do not always become native languages, but they sometimes do. The key is that Aramaic was considered the prestigious language, and speakers of other languages needed to learn and use it, and over time this extended to their native language as well. For more information on this process, if you're interested, /r/linguistics might be a good place to go.
In terms of when, it was a slow process, over centuries. It's important to note that Hebrew and Aramaic are similar, so you can to an extent blend them, which makes the change less radical. When most biblical texts were being written, Hebrew was the language medium, and it seems Hebrew was the common language. But even within the bible there are large sections of Aramaic in the later books. It seems that by the time the Second Temple was destroyed, there were very few, if any native Hebrew speakers left, even though many were still conversant in it. And certainly their Hebrew had been affected by Aramaic, effects which lasted in Hebrew thereafter.