Benjamin (בִּנְיָמִון) was originally named Ben-Oni (בֶּן־אוֹנִי). Rachel died giving birth, and named him "Son of My Sorrow," but his father preferred "Son of the Right Hand." This signifies a position of favour and authority, which we find to be the case when his father refuses to let him go to Egypt with his brothers, and we all know the story. What is interesting to me is that the only times I can think of in all of Scripture that a man is mentioned as being right or left handed (Ehud in Judges 3:15 and the slingers of Judges 20 and 1 Chronicles 12:2), he is a left handed "son of the right hand."
In the book of Judges, Israel's rise to power in the Middle East is being traced. They begin faithfully and obediently, and serve God all the time of Joshua's life (110 years, one ten less than Moses' 120, which is the perfect government of ten twelves). They rebel, and are delivered into the hand of "Chushan-rishathayim," literally "Kuwshan of double-wickedness" for eight years, and from whom they are saved by Othni-el the "force of God," son of "the hunter," nephew of Caleb. Now we hit verse twelve of chapter three, and it gets interesting. God strengthens the Calf (Eglon), king of "her father" (Mo-ab) against Israel, who collects Lot and Esau and takes the "erect" city, the city of the palm trees; then the Israelites return to Egypt in all but geography, becoming slaves to their gods. But upon their repentance, God raises up Unity (Ehud), the left handed son of the right hand to kill Eglon and deliver Israel after three sets of six years: three weeks of years with no Sabbaths (Jordan stretched you a little, but I shall stretch you much...).
So, here is my theory.
The right hand is a place of honour. But Israel throughout the book of Judges is dishonourable: they rebel time after time, and the book is a chiasm with the theme of left-handedness being one of the first bookends (chapter three, immediately after the introduction, and chapter 20, immediately before the conclusion; in each case, Judah goes up first, in each case Israel is unified: Ehud means unity, and many more similarities). Now this is in itself significant, but it's made far more so by the fact that Hebrew has no word for left-handed. The Hebrew actually reads that Ehud was a man whose "right hand was bound" (הַיְמִינִ֔י אִ֥ישׁ אִטֵּ֖ר יַד־ יְמִינֹ֑ו) and the seven hundred slingers were "men of bound right hands." So in Judges, Israel's honour is bound: they are rebellious and dishonourable. But then they unify against this dishonour and destroy it in submission to God.
But right after this (chronologically), they again rebel: they have been dishonoured among the surrounding nations because of rebellion, and only freed by God's mercy through judges. But now they want to be just like the surrounding nations, not a lesser nation to be conquered and re-conquered. So they are given a king of the left hand, whose dishonour shapes the noblest king of Israel, whose dynasty takes Israel from her dishonour into her greatest glory.