This phrase seems to have deep roots in Christianity. It feels as a modern to emphasize the level of sacrifice - only son vs one of many. Is there more historical weight or meaning behind this phrase in 1st century Judea (or more likely some later point when it entered use in the church?) Is the emphasis on the loss of the only child or the fact that Jesus IS the only child?
Like many biblical questions, it is partially a translation quirk.
If you trace that particular phrase (from John 3:16) closer to it's Greek origins you arrive at the word "monogenes". This word has a few meanings but in this context is something something closer to "unique in kind", and was probably used to highlight the unique position of Jesus as sharing Gods divine nature (as opposed to believers who might be called "Gods Children" as per Ephesians).
Interestingly the phrase "only begotten son" also appears in the bible to refer to someone other than Jesus. Specifically Hebrews uses the term to refer to Issac as Abraham's only begotten son. This is meaningful for your question because Issac wasn't Abraham's only son (he had 8 in total), but Issac was unique in that he was Abraham's only son with Sarah.