Jeroboam and Rehoboam maybe sound similar but have completely diffrent meanings; in fact they might the exact opposites of each other.
They are both compound words and they both have the same ending עָם, which means nation, but the beginning letters is what drastically changes their meanings. It's not entirely clear the etymology of their names and the bible doesn't tell us, but it should be fairly easy to reconstruct their names and what they stand for.
The root ירב can either mean multiply (see Exodus 1:7) or fight/contend (see Judges 6:32). According to this the name should be interpreted as "whose people is numerous", and should be pronounced יָרֹב עָם. I believe the root ירב should be vowelized יָרֶב which means fight/contend, similar to the name יְרֻבַּעַל which the book of Judges clearly links with יָרֶב explaining it thus: "Let Baal contend with him." You can see more on this here. This meaning is indeed very appropriate here, Jeroboam has caused a rebellion among the people against Judah and its king, and was thus aptly nicknamed (by the Judeans) "the one who contends with his nation", or "let the nation contend with him". Whether this was the original meaning of his name or a new meaning that was later added is not entirely clear, but the point is that people came to see him as a rebel, one who caused a rift in the nation of Israel, as his nickname strongly suggests.
Rehoboam has a more positive undertone. According to this it comes from the root רָחָב which means elsewhere in the bible freedom or making room. The Judeans were full of hope that their king Rehoboam will restore their freedom as opposed to Jeroboam who has created a rift between them and the rest of the nation. Although in the end their hopes were dashed when he answered them ‘My little finger is thicker than my father’s waist. 11 My father laid on you a heavy yoke; I will make it even heavier. My father scourged you with whips; I will scourge you with scorpions.’ (1 Kings 12:10-12).
So to summarize, it seems that with the parallel names the biblical author wishes to highlight the contrast between the good kings of Judah and evil kings Israel. The name "Jeroboam" symbolizes his callousness and the rift he has caused, whereas "Rehoboam" symbolizes the times of freedom that he has ushered (or was supposed to usher) unto the land.
As for Abijah, they are indeed the same name, but I cannot see any significance in this coincidence, but it is very likely that there is something at play here.
Hope this is of any help to you.