Shortest answer is Plutarch (c. 46 AD – 120 AD), Roman citizen and Greek essayst and biographer. Longer answer follows. Although Solon wrote himself - he was a poet as well as a lawmaker for instance - his surviving works are poorly fragmented. His works are known from the authors such as Aristotle (Athenian Constitution, Politics) and his life was documented by the likes of the ancient authors Herodotus and Plutarch. Although they lived later than him (Solon died around 558 BC, Herodotus was born c. 484 BC and Plutarch c. 46 AD) they are still our main source of informations. Herodotus mentions him in The Histories (440 BC), and Plutarch offers Solon's biography in the Lives of the Noble Greeks and Romans (or simply Parallel Lives or Plutarch's Lives). The second one is interesting for us and will offer our answer. Plutarch, The Life of Solon, 78‑97, 6:

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"On his visit to Thales at Miletus, Solon is said to have expressed astonishment that his host was wholly indifferent to marriage and the getting of children. At the time Thales made no answer, but a few days afterwards he contrived to have a stranger say that he was just arrived after a ten days' journey from Athens. When Solon asked what news there was at Athens, the man, who was under instructions what to say, answered: "None other than the funeral of a young man, who was followed to the grave by the whole city. [...] The man said it was [Solon's son that is dead]; whereupon Solon began to beat his head and to do and say everything else that betokens a transport of grief. But Thales took him by the hand and said, with a smile, "This it is, O Solon, which keeps me from marriage and the getting of children; it overwhelms even thee, who art the most stout-hearted of men. But be not dismayed at this story, for it is not true." Such, at any rate, according to Hermippus, is the story of Pataecus, who used to boast that he had Aesop's soul."

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So, Plutarch took this story from old Athenian writer (worked sometimes in 5 century BC), Hermippus, fierce rival of Pericles, and author of poems and comedies of which only fragments of nine are preserved, as well as their titles. Arnaldo Momigliano in his The Development of Greek Biography (Harvard, 1993) even arguments that Plutarch summarized work of Hermippus through The life of Solon. However, that is not important at this point. Anyway, what happened later with Solon's son, we do not learn, but the fact that story about Solon's son's death is not true. Plutarch took this opportunity to convey a lesson: one whole next paragraph is dedicated to Plutarch's opinion of Solon's behaviour:

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"However, it is irrational and ignoble to renounce the acquisition of what we want for fear of losing it; for on this principle a man cannot be gratified by the possession of wealth, or honour, or wisdom, for fear he may be deprived of them. Indeed, even virtue, the most valuable and pleasing possession in the world, is often banished by sickness and drugs. [...] For it is weakness, not kindness, that brings men into endless pains and terrors when they are not trained by reason to endure the assaults of fortune. Such men do not even enjoy what they long for when they get it, but are filled with continual pangs, tremors, and struggles by the fear of future loss. However, we must be fortified not by poverty against deprivation of worldly goods, nor by friendlessness against loss of friends, nor by childlessness against death of children, but by reason against all adversities. This, under present circumstances, is more than enough on this head."

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Here we can go back to Bacon.

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"Solon when he wept for his son's death, and one said to him, 'Weeping will not help;' answered, 'Alas, therefore I weep, because weeping will not help'."

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If we accept Plutarch's story as only possible resource Bacon could derive his story, and that's definetly the earliest and most extensive account we can get our hands on, his Apophthegm can be observed as a romanticized summary of Plutarch's two chapters. Furthermore, word apophthegm means a short, witty, instructive saying; an aphorism or maxim. And that's precisely what this is. A witty aphorism, maybe nice example of petitio principii – a type of circular reasoning and an informal fallacy which often occurs in an indirect way such that the fallacy's presence is hidden, or at least not easily apparent. Furthermore, to conclude connection with Bacon and Plutarch, we can offer the sole beginning of Bacon's Apophthegms (A.D. 1625) where he mentions Plutarch:

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"Julius Cæsar did write a collection of apophthegms, as appears in an epistle of Cicero; I need say no more for the worth of a writing of that nature. It is pity his book is lost: for I imagine they were collected with judgment and choice; whereas that of Plutarch and Stobæu, and much more the modern ones, draw much of the dregs. Certainly they are of excellent use."

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Literature:

Plutarch, Lives of the noble Grecians and Romans

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The Works of Francis Bacon, Volume 1/Apophthegms

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Arnaldo Momigliano, The Development of Greek Biography, Harvard, 1993

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Philip A .Stadter, Plutarch and His Roman Readers, 2015

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