A basic principle to understanding Bible is to try to grasp the significance behind of a ‘string’ of identical terms, even if they are scattered across the Bible.
Granted, the usage of the same specific term in every possible context must be viewed through the context lens itself, but – in the same time – a specific and identical term must possess a semanthic thread which links all the scattered occurrences of it.
If no, why the writer did choose to utilize the same term when he had the possibility to choose different (and sometimes more specific) terms?
So, we have to ask ourselves another question, What is the semanthic thread that links all the Bible texts like Jon 4:10, 2 Sam 12:5, Deu 25:2; Psa 79:11, and Pro 31:8 (these latter four passages have been quoted by Abu Munir ibn Ibrahim)?
In a number of languages, including Hebrew, English, and Italian, the term ‘fruit’ (בן [BN] in Jon 4:10) also stands for the effect or consequence of something else, the natural consequence of a process, as in the sentence “those were the fruits of his labour”. The usage of this idiom (‘son of…’) exhorts the readers to imagine the final state/condition of the individual at issue, as a result of a consequential process, not merely an accidental event, coming unexpectedly on the subject.
Accordingly, in Psa 127:3 we find a clear parallelism that makes light to the question at issue, namely ‘fruit (of the belly)’ = ‘sons’.
Sure, a possible translation of ישׂראל בני (BNI ISRAL) is ‘Israelites’ (in view of the literal ‘sons of Israel’ rendering), but in this case we have to be aware that so we lose the immediate connection with the basic illustrated above concept, in this specific case, the natural consequence of the procreative process triggered by Jacob (Israel). For this reason, I would prefer to translate this expression with “sons of Israel”, or, “progeny of Israel”. (Exo 1:5 gets even a more articulate manner to express the same concept, namely, “all the souls coming forth the thigh [euphemism for ‘penis’] of Jacob”)
See how this mode can be applied to the passages mentioned by Abu Munir.
2 Sam 12:5: “he will become a natural consequence [lit. ‘son’] of Death”, namely, as we have habit to express ourselves > “he must be executed by a violent death”, or, more simply > “he must be put to death” (the same reasoning line can be applied to Psa 79:11).
Note how every time you pass through the symbol ‘>’ (it indicates here a slight semantic deviation of understanding, made with the aim to ‘simplify’ a more articulate concept) you lose some information, inevitably.
In fact, if we translate that sentence as ‘he must be put to death’, we forfeit - on the way – the connection with the natural consequence of the process (in this case, of Nathan-mentioned man’s behavior), where, in the Bible is expressed in various wordings (“For they have been planting the wind, and their fruit will be the storm” [Hos 8:7, BBE]; “You have been ploughing sin, you have got in a store of evil, the fruit of deceit has been your food” [Hos 10:13, BBE]; “According as I have seen, they that plow iniquity, and sow mischief, reap the same” [Job 4:8, JPS]; “He that soweth iniquity shall reap vanity” [Pro 22:8, JPS], along with many other Bible passages…
In Deu 25:2, the literal “he is son of a smite” can be traslated not merely “is to be smitten” (as Young renders, as well as others), but – better – “he will become a natural consequence [lit. ‘son’] of Stroking” (πληγη in LXX).
In Pro 31:8, the literal rendering “sons of the Passing-Away” (commonly translated by “[those] appointed to destruction”), can be better translated “those that are becoming the natural consequences [lit. ‘sons’] of the Passing-Away”, or “…the Change [from life to death].”
About the translations presented above, someone may say: “But these translation are wooden, somewhat stiff!” Sure, admittedly they are right, from the viewpoint of a Western-cultured peoples. But the Bible is not a fruit (!) of the Western culture, at all!
Sure, we (translators) have to adapt some language structures in some comprehensible ways, but avoiding to hyper-simplify the concepts, namely, reaching the stage where to the reader remains an amount of information lower than the original text.
This kind of translation (I've presented) have the undeniable advantage to transmit the nuance the original author did wish to convey his readers.
Now, if it is preferable obtain a smoother – but less precise – rendering, instead of a translation more precise, but more wooden, this will be a translator’s choice, that will depend on the goal he put in front of him…
As regards Gen 7:6 (mentioned this time by Broberg), the reasoning line is similar. The corresponding Bible Hebrew expression of the today-style (for an example, drawn by a back translation of Luk 8:42) “was 12 years old” can be translated – aptly -היתה שׁנה עשׂרה שׁתים, without any mention of a given ‘son’ (בן).
So, since the Genesis’ author did choose to render that sentence utilizing the term ‘son’ (בן), we have to conclude he (Moses, traditionally) did want stress – above the mere count of Noah’s years of life – the sense that the patriarch in that epoch was also the 'fruit' of the natural consequences of his 600-years of life's experiences.
In conclusion, to answer directly to the Jon 4:10-question made (“What is the actual meaning of this phrase?”):
A better translation is: “as a natural consequence [lit. ‘son’] of one-night growth […] as a natural consequence [lit. ‘son’] of one-night decay”.
According this translating mode the meaning of this Bible passage is so easily grasped.
I hope these notes will be useful to you.