Short answer: It combines the name of God with a pun on the word for pain or grief.
According to the 13th c. Jewish commentator David Kimhi (Radak), the name results from a Hebrew wordplay involving a transposition of letters. The pun relates to the idea of harm or grief. Jabez caused his mother pain/grief in childbearing. In naming him, she says "I gave birth to him in pain," echoing God's curse upon Eve:
Genesis 3:16
To the woman he said: I will intensify your toil in childbearing; in
pain you shall bring forth children.
In 1 Chron. 10b, the narrator records Jabez' prayer to be protected from pain:
"Make me free of misfortune, without pain!” And God granted his
prayer.
Etymologically, the problem is that the word for grief/pain is astab - עָצַב while the name given to him by his mother is simply ya‘·bêṣ- יַעְבֵּ֔ץ. So the two words, which are quite dissimilar as we hear them today, do share two consonants and a probable vowel. Kimhi writes:
יעבץ (Jabez)/ Considering the story, this involves a transposition [of
letters]—for according to the story it should have said יעצב.
Hebrew-speakers are not particular on this.
For Radak, the words are close enough to each other to understand the pun. Readers might also recall Gen. 3:16, in which the "b" sound is further emphasized (bə·‘e·ṣeḇ - in pain). Another element in this puzzle is the syllable "ya/ja" added to Jabez' name, which introduces God's name (yahweh) into the equation. This makes the pun more understandable.
Conclusion: Jabez or "Ya-bes" combines a play on the word for pain with the name of God. It means something like "grief/pain from God." Whether original readers would have understood the pun is debatable.