I've noticed that the third person aorist form of γίνομαι (namely ἐγένετο) is never an action or a dynamic verb, like ἦλθεν. So it's range is virtually identical to that of ἦν. The only difference is that ἐγένετο has the sense "came to be" while ἦν means simply "was." In some contexts the two senses are interchangeable, but obviously in others they are not. In John 1:1b the two would have the same basic meaning, since ἦν by itself does not connote eternity, so that at some point in the past the Word had to have come to be with God. Here are some examples of ἐγένετο..
Here's a sampler:
Καὶ ἐγένετο ὅτε ἐτέλεσεν ὁ Ἰησοῦς τοὺς λόγους τούτους, ἐξεπλήσσοντο οἱ ὄχλοι ἐπὶ τῇ διδαχῇ αὐτοῦ·
Matt. 7:22
Καὶ ἐγένετο ἐν ἐκείναις ταῖς ἡμέραις ἦλθεν Ἰησοῦς ἀπὸ Ναζαρὲτ τῆς Γαλιλαίας καὶ ἐβαπτίσθη εἰς τὸν Ἰορδάνην ὑπὸ Ἰωάνου
Mark 1:9
.Καὶ ὅτε ἐγένετο ἐτῶν δώδεκα, ἀναβαινόντων αὐτῶν κατὰ τὸ ἔθος τῆς ἑορτῆς,
Luke 2:42
προσελθόντες δὲ διήγειραν αὐτὸν λέγοντες Ἐπιστάτα ἐπιστάτα, ἀπολλύμεθα. ὁ δὲ διεγερθεὶς ἐπετίμησεν τῷ ἀνέμῳ καὶ τῷ κλύδωνι τοῦ ὕδατος· καὶ ἐπαύσαντο, καὶ ἐγένετο γαλήνη.
Luke 8:24
πάντα δι’ αὐτοῦ ἐγένετο, καὶ χωρὶς αὐτοῦ ἐγένετο οὐδὲ ἕν
John 1:3
Ἐγένετο ἄνθρωπος, ἀπεσταλμένος παρὰ Θεοῦ, ὄνομα αὐτῷ Ἰωάνης·
John 1:6
ἐν τῷ κόσμῳ ἦν, καὶ ὁ κόσμος δι’ αὐτοῦ ἐγένετο, καὶ ὁ κόσμος αὐτὸν οὐκ ἔγνω.
John 1:10
Καὶ ὁ Λόγος σὰρξ ἐγένετο καὶ ἐσκήνωσεν ἐν ἡμῖν, καὶ ἐθεασάμεθα τὴν δόξαν αὐτοῦ, δόξαν ὡς μονογενοῦς παρὰ Πατρός, πλήρης χάριτος καὶ ἀληθείας.
John 1:14
ὅτι ὁ νόμος διὰ Μωυσέως ἐδόθη, ἡ χάρις καὶ ἡ ἀλήθεια διὰ Ἰησοῦ Χριστοῦ ἐγένετο.
John 1:17
Ταῦτα ἐν Βηθανίᾳ ἐγένετο πέραν τοῦ Ἰορδάνου, ὅπου ἦν ὁ Ἰωάνης βαπτίζων.
John 1:28
καὶ ἐγένετο ἄφνω ἐκ τοῦ οὐρανοῦ ἦχος ὥσπερ φερομένης πνοῆς βιαίας καὶ ἐπλήρωσεν ὅλον τὸν οἶκον οὗ ἦσαν καθήμενοι,
Acts 2:2
Καὶ τῷ ἀγγέλῳ τῆς ἐν Σμύρνῃ ἐκκλησίας γράψον Τάδε λέγει ὁ πρῶτος καὶ ὁ ἔσχατος, ὃς ἐγένετο νεκρὸς καὶ ἔζησεν
Rev. 2:8
Καὶ εἶδον ὅτε ἤνοιξεν τὴν σφραγῖδα τὴν ἕκτην, καὶ σεισμὸς μέγας ἐγένετο, καὶ ὁ ἥλιος ἐγένετο μέλας ὡς σάκκος τρίχινος, καὶ ἡ σελήνη ὅλη ἐγένετο ὡς αἷμα,
Rev. 6:12
etc. etc.
Best wishes to all,