The name Jabez in the Bible
In the Bible there is one man and one city named Jabez, both of Judah, and the city is probably named after or even by the man. The man shows up in 1 Chronicles 4:9-10 in a two verse novel that suggests he was a prominent thinker and leader in his time. It's unclear who his parents are, but his (unnamed) mother calls him Jabez, saying, "Because I bore him with pain," which is curious because every child birth is painful (Genesis 3:16). But perhaps she wasn't referring to child birth, and to mind comes what Simeon tells Jesus' mother Mary, "and a sword will pierce even your own soul" (Luke 2:35).
The town Jabez is mentioned just prior to Jabez the man, in 1 Chronicles 2:55, where Jabez is said to be peopled by families of scribes: Tirathites, Shimeathites and Sucathites. Obviously, Jabez was a center of learning, which brings to mind what Qoheleth writes: "In much wisdom there is much grief, and increasing knowledge results in increasing pain." (Ecclesiastes 1:18).
Etymology of the name Jabez
After all these references to pain, we would expect that the name Jabez would clearly have to do with it, but no. The name Jabez is of unknown origin and we have no idea what the mother of Jabez is talking about. BDB Theological Dictionary even declares a whole root for this name (עבץ), with no known meaning, no cognates, and no other derivatives than the name Jabez.
However, the noun which Jabez' mother uses to say that she bore him in pain comes from the verb עצב ('asab). Note that this verb has the same letters as the name Jabez, and that's probably no coincidence. It sometimes happens that in names letters are deliberately swapped (this is called metathesis), but what the literary meaning of this might be is not clear:
Even though there are some compelling theories, we'll probably never know why Jabez' mother turned to metathesis in naming her son. But what is sure is that she's making a very profound statement about knowledge on one hand and the process of being formed into the likeness of the Lord on the other.
For a meaning of the name Jabez, NOBSE Study Bible Name List reads He Makes Sorrowful. Jones' Dictionary of Old Testament Proper Names proposes He Will Cause Pain. As stated above, BDB Theological Dictionary declares meaning unknown.