According to my copy of the Evev-Shushan dictionary, the word Biryonim is from the root (Bar), although this is listed as only a possibility.
The source for the word is from the Babylonian Talmud, Gittin 46a. They were a Jewish military group at the time of the destruction of the second Temple, who fought the Romans and the Jews who wanted to try to resolve the war peacefully.
Today, it refers to gangs of troublemakers.

Incidently, "Baryonim" is a composite metal used in manufactuting, with a high barium content.