Here are some examples of γίνομαι (the aorist middle form ἐγένετο is employed in John 1:14) and brief comments about how it is utilized in each cited verse:
Matt. 4:3-used by Satan who asks Jesus to make some stones become loaves of bread: καὶ προσελθὼν ὁ πειράζων εἶπεν αὐτῷ Εἰ Υἱὸς εἶ τοῦ Θεοῦ, εἰπὲ ἵνα οἱ λίθοι οὗτοι ἄρτοι γένωνται.
John 1:3-The Apostle writes that all things (πάντα) came into being through the LOGOS (δι’ αὐτοῦ ἐγένετο).
John 1:12-Humans who exercise faith in Jesus and receive him are given the authority "to become" (γενέσθαι) God's children (τέκνα Θεοῦ ).
John 1:14-The LOGOS became flesh: Καὶ ὁ Λόγος σὰρξ ἐγένετο καὶ ἐσκήνωσεν ἐν ἡμῖν, καὶ ἐθεασάμεθα τὴν δόξαν αὐτοῦ, δόξαν ὡς μονογενοῦς παρὰ Πατρός, πλήρης χάριτος καὶ ἀληθείας.
Heb. 11:3-τοὺς αἰῶνας came to be out of things that do not appear: Πίστει νοοῦμεν κατηρτίσθαι τοὺς αἰῶνας ῥήματι Θεοῦ, εἰς τὸ μὴ ἐκ φαινομένων τὸ βλεπόμενον γεγονέναι.
Cf. Matt. 5:45 and countless other examples.
Image from BDAG.