And so it was, as her soul was departing (for she died), that she called his name Ben-Oni; but his father called him Benjamin. Genesis 35:18 (NKJV)
It's often stated that "Ben-Oni" means "son of my mourning" or "son of my sorrow," which is a strange thing for a woman to name her son. It would seem appropriate, then, for Jacob to want to rename his son to something stronger, or at least more positive.
However, footnotes of the Hebrew meaning of "Ben-Oni" point out that the name can either mean "son of my mourning" or... something else entirely: "son of my strength."
Here, the root word "oni" points to mourning:
They shall not offer wine offerings to the Lord,
Nor shall their sacrifices be pleasing to Him.
It shall be like bread of mourners (אוֹנִים) to them; Hosea 9:4a (NKJV)
But here, the meaning is strength:
“Reuben, you are my firstborn,
My might and the beginning of my strength (אוֹנִי),
The excellency of dignity and the excellency of power. Genesis 49:3 (NKJV)
It's from this dual usage of the word that some rabbis say that the naming of Benjamin was like this: Rachel, knowing that the childbirth would end her life, intended to attach the idea of mourning to his name. Jacob, wanting his son to carry the name his mother gave him (all of Jacob's children were named by Rachel or Leah), chose to honor the "strength" meaning of the name when he renamed him "Benjamin." The meaning "son of my right hand" implies "son of my strength."