This is a mere translation problem. The King James Bible is gives a confusing translation for this text, reading:
And it came to pass, as her soul was in departing, (for she died) that she called his name Benoni: but his father called him Benjamin.
Whereas the New International Version reads:
“As she breathed her last—for she was dying—she named her son Ben-Oni. But his father named him Benjamin.”
Now it it true that the underlying Hebrew reads “nap̄·šāh” (herself,her soul) and “bə·ṣêṯ” (went out, went forth, was departing, had gone, was gone, came out, etc.), but it is obvious that the author of Genesis, Moses, is writing euphemistically. We often do the same thing: “he passed away,” “he passed,”.
The New Living Translation reads:
Rachel was about to die, but with her last breath she named the baby Ben-oni (which means “son of my sorrow”). The baby’s father, however, called him Benjamin (which means “son of my right hand”).
Everyone in our English-speaking American culture understands that all of these expressions are euphemisms for “he died.” We do not like to use the harsher phrase “he died,” especially around the loved ones of the “dearly departed,” so we use euphemisms (from the Greek eúphēmos, meaning “sounding good”).
*What Moses is telling us is that Rachel was dying, and with her last breath she named her son Ben-Oni (“son of my sorrow”)."
F.D. Nichol