the idea of anything being reborn goes against orthodox early Buddhist teachings
In this later comment, yuttadhammo wrote,
the pali word is "upapajjati", which means something like "he arises". No sense of "re-" at all
So that's the distinction that he's trying to make, in the context of that question:
OP is implying that something that has ceased might arise again. This doesn't happen. Talking about rebirth as a theory is fine, as long as we understand that in reality nothing is actually "reborn". It's a poor name for what really happens.
If I look in the Pali dictionary for the word punabbhava cited in Thiago's answer, the dictionary definition is "renewed existence, new birth" rather than "rebirth".
Looking at the root of the word Puna it suggests meanings like "again" but also "tail", "poop", "behind".
That reminds me of this definition of the word akalika (which means 'timeless'), which includes the following paragraph:
It is probably most useful here to go to the ultimate roots: a = no; ka = shit; li = line; ka = shit. The track of scat left by an animal. The hunter sees: This is the track a week old, this is only two days old, this is from yesterday, and here it is now, eating. That would point to the original meaning of the term to be closer to successive than to simultaneity.
So by analogy, "born again" or "living again" might be like "pooping again": i.e. it is "again", "re-pooping", but let's be clear that they're two different poops.
where did the concept of "rebirth" come from
Here are a couple of articles from accesstoinsight:
The Truth of Rebirth starts with,
Rebirth has always been a central teaching in the Buddhist tradition.
This article includes hyperlinks to the suttas.
Does Rebirth Make Sense? doesn't include specific hyperlinks but does say things like,
The teaching of rebirth crops up almost everywhere in the Canon, and is so closely bound to a host of other doctrines that to remove it would virtually reduce the Dhamma to tatters. Moreover, when the suttas speak about rebirth into the five realms — the hells, the animal world, the spirit realm, the human world, and the heavens — they never hint that these terms are meant symbolically. To the contrary, they even say that rebirth occurs "with the breakup of the body, after death," which clearly implies they intend the idea of rebirth to be taken quite literally.
Why is it a topic of discussion?
It's a topic of discussion because if a doctrine of "rebirth" implies there's something (perhaps a permanent self or soul) that "is reborn", then that seems to contradict other Buddhist doctrines.