Euripidis Tragoediae (Nauck 3rd ed.)

...leisure and to pass his time in the study of poetry. Although his father is reported to have immigrated to Athens from Boeotia 8, Euripides was nevertheless an Athenian, inscribed in the deme of the Phlyenses 9: he is said, however, to have been born on the island of Salamis, to which it is probable his parents fled because of the imminent danger from the Persians 10. He spent his boyhood and perhaps his youth in the same place: for this, and nothing more, I would conclude from the passage of Gellius N.A. 15.20.5: Philochorus reports that there is a grim and horrible cave on the island of Salamis, in which he used to write 11. At first, he was perhaps brought to the wrestling school palaestra unwillingly, not so much by his own will as by his father's command: certainly, he did not favor the exercises and character of athletes afterward 12, and it is reported that his father was moved by an oracle's response to command his son to become an athlete 13. He is also reported to have given attention to painting 14.

8) See the passage of Nicolaus Damascenus cited in note 6.
9) Harpocr. p. 182.5 Phot. Lex. p. 651.21 Etym. M. p. 795.39 Suid.: Phlyea (Phlyenses Harp.) is a deme of the Cecropid tribe; the tragic poet Euripides was from this deme. Elmsley suspected that Euripides was a Cholleides from Ar. Ach. 406.
10) Vita Eur.: Athenian indeed, but he was born in Salamis. Hence he is called Salaminian, Corp. Inscr. 6052 (see note 5). Suidas: Euripides, son of Mnesarchus or Mnesarchides, who fled and moved to Boeotia: then to Attica. It seems due to the ingenuity of grammarians that Priscian vol. 1 p. 63.3 ed. Hertz reports that Euripides was named after the Euripus.
11) Cf. life of Eur. vs. 61: They say that having constructed a cave in Salamis, which had an opening to the sea, he spent his days there, fleeing the crowd.
12) In particular see fr. 284: For among the ten thousand evils in Greece, there is nothing worse than the race of athletes. Who first neither learn to live well, nor could they, etc.
13) Vita Eur.: He practiced at first the pankration or boxing, his father having received an oracle that he would win crowned contests; and they say he won at Athens. Gellius 15.20.2 embellished these with more detail: But to his father, when he was born, it was answered by Chaldeans that this boy, when he had grown up, would be a victor in contests; that this was fated for the boy. The father interpreted that he ought to be an athlete, and having strengthened and exercised his son's body, he led him to Olympia to compete among the boy athletes. And at first, he was not received into the contest due to his ambiguous age, but afterward he fought in the Eleusinian and Thesean contest and was crowned. Nor was there lacking one who would report the oracle itself given to the father. cf. Oenomaus in Eusebius P. E. V p. 227 C: A son will be yours, Mnesarchides, whom all men will honor, and he will rush toward noble glory, and will put on the sweet grace of sacred crowns.