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Beor 
The name Beor is applied twice in the Bible. First as the name of the father of Bela, the first king of Edom (Gen 36:32), which lies due south of the Dead Sea. Secondly as the father of Balaam of Pethor (Num 22:5), which is in Aram Naharayim (Deut 23:4), which means Aram of the Two Rivers. This is traditionally interpreted to be Mesopotamia, but more recent research suggests the Jordan/Jabbok region. See the discussion under Pethor.
Because some consider Pethor to be located in Canaan, it becomes rather tempting to suggest that the Beor/Bela couple of Edom is in fact the same as the Beor/Balaam couple of Pethor:
In the Edom king-list of Gen 36 we see kings being imported from all over the place (even from, what do you know, the Euphrates - 36:37). However, in Num 20:14 Edom is already a kingdom. The king of Edom refuses Israel passage, and even comes out with an armed force against Israel. It's highly unlikely that the nerdy, donkey-riding Balaam of two chapters later is the same as Edom's valiant first king Bela, who stayed king onto his death (v33).
The name Beor looks like it has to do with any of the roots  (ba'ar 263-264.1):
(ba'ar 263) means to burn, consume, but most often used in a figurative sense to describe anger, passion, intrigue, etc. In Hebrew there are a few verbs and nouns that describe fire, but ba'ar "stresses the consuming and contagious qualities of fire especially in the religious context" (TWOTOT ). The noun  (be'era 263a), fire, occurs only once, and that in close proximity to the verb: Ex 22:6, "...he who kindled the fire shall repay."
(ba'ar 264) means to be brutish (Jer 10:8, Isa 19:11). An obvious derivation is the noun (ba'ar 263b), a brute, brutish person. BDB lists  (be'ir 264a), beasts, cattle under ba'ar 263, but TWOTOT lists it under ba'ar 264. Both, however, emphasize that this word has nothing to do with the rare usage of the verb ba'ar 263 in the sense of some animal 'grazing' someone's field (Ex 22:4) the way a fire would (note the same verb meaning burn in v6). TWOTOT declares this word 'simply' an alternative synonym for  (behema 208a) or  (miqneh 2039b), both meaning cattle. BDB lists equivalents in cognate languages, apparently to suggest that (be'ir 264a) was not derived of root but rather grafted upon it.
(ba'ar 264.1) was, until recently, considered part of (ba'ar 263) but lately it is viewed separately (or so sayeth TWOTOT ). This is, of course, tremendously clever, but one may wonder if the Hebrews themselves knew about this. This completely separate root (ba'ar 264.1) occurs twenty-seven times and mostly denotes a removal of evil from the land. That this removal became typically depicted as purification by fire has apparently no bearing on the discussion of where 264.1 came from.
The name Beor may mean any of the above. BDB and NOBSE both read A Burning.
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