γενέσθαι in the Fourth Gospel - Part 1
John uses the specific γενέσθαι in contexts which describe change. In addition to 8:58, this aorist infinitive form is used in 1:12, 3:9, 5:6, 9:27, 13:19, and 14:29. The first use, which is also the focal point of the chiasmus of the Prologue and one of the most significant statements in the Fourth Gospel, paints a clear picture of John's meaning:
12 But to all who did receive him, who believed in his name, he gave the right to become, γενέσθαι, children of God, 13 who were born, not of blood nor of the will of the flesh nor of the will of man, but of God. (ESV)
Because the specific change is described, there is no ambiguity in meaning. Individuals who previously were described one way become children of God. Not only was, or is, anyone who makes this choice changed, the change is not gradual; rather there is a specific point in time at which the individual the change occurred:

As illustrated above, the element of time is not a "generic" before and after. Time looks back to the moment of birth and forward without end.
After the saying in question, John again uses γενέσθαι including the temporal condition:
26 They said to him, “What did he do to you? How did he open your eyes?” 27 He answered them, “I have told you already, and you would not listen. Why do you want to hear it again? Do you also want to become, γενέσθαι, his disciples?” 28 And they reviled him, saying, “You are his disciple, but we are disciples of Moses. 29 We know that God has spoken to Moses, but as for this man, we do not know where he comes from.” (John 9)
The Pharisees do not want to become disciples of Jesus. The element of time is essential for their unwillingness to change: God has spoken to Moses, but they do not know where Jesus comes from. As with 1:12, there is no ambiguity in the meaning of the aorist infinitive. Their current state is based on the past and they will not become; they will remain disciples of Moses.
While it is true in general the aorist infinitive may be ambiguous, in both 1:12 and 9:27, there is no ambiguity: γενέσθαι describes a before and after condition. In addition, if there is a potential for ambiguity, it is in the implied beginning and end of time. That is, when did the Pharisees become disciples of Moses and, if they had decided to become Jesus' disciples, how long would they remain (cf. John 6:66). On the other hand, 1:12 clearly identifies the point of beginning, birth, and leaves the future for the reader to decide. That is, if the rebirth is permanent and children remain so forever, then, as the illustration shows, there is no ending point.
γενέσθαι in the Fourth Gospel - Part 2
In addition to 1:12 and 9:27, the other uses of γενέσθαι (3:9, 5:6, 13:19, 14:29) also involve change, and like 1:12 and 9:27, all place that change in context to another event in time. As in John 8:58, two use preface γενέσθα with the adverb, πρὶν - "before:"
The official said to him, “Sir, come down before my child dies.” (4:49)
λέγει πρὸς αὐτὸν ὁ βασιλικός κύριε κατάβηθι πρὶν ἀποθανεῖν τὸ παιδίον μου
And now I have told you before it takes place, so that when it does take place you may believe. (14:29)
καὶ νῦν εἴρηκα ὑμῖν πρὶν γενέσθαι ἵνα ὅταν γένηται πιστεύσητε
The statement in 14:29 combines both πρὶν and γενέσθαι and is literally before becomes. Here Jesus told the disciples they would receive the Holy Spirit before the event, so that when it happened, they would believe. Again, because the context includes a second event, there is no ambiguity in John's use.
Conclusion - Part 1
The primary question is: outside of John 8:58, how is the term γενέσθαι used by the author John when it comes to timing of what the verb is describing? The answer is outside of 8:58, John always uses γενέσθαι to describe a change which is never gradual but occurs at a specific point in time. In addition John always places γενέσθαι in context which includes a past time and conveys on-going time after the change.1
John 8:58
Like all other uses of γενέσθαι, John 8:58 includes another event:
56 Your father Abraham rejoiced that he would see my day. He saw it and was glad.” 57 So the Jews said to him, “You are not yet fifty years old, and have you seen Abraham?” 58Jesus said to them, “Truly, truly, I say to you, before Abraham γενέσθαι, I am."
(John 8)
In context, before Abraham is responding to the Jews misrepresenting the past event. They recognize Jesus is referring to the past, but reject the possibility Jesus, who is not old enough, could have seen Abraham.2
Jesus responds beginning with a correction to their attempt to reverse His temporal relationship with Abraham. His response could be paraphrased, "do not twist My words around! Before Abraham saw My day... The Jews had tried to put place Jesus after Abraham, but Jesus says it was Abraham who rejoiced to see Jesus' day.
Jesus further closes off any possibility of misrepresenting the past event by adding "I am" which introduces another temporal when combined with γενέσθαι. I am in context, that is before Abraham becomes can be illustrated:

Jesus response could be paraphrased: Do not twist My words around. Abraham rejoiced to see My day but/and before Abraham becomes, I am."
The logical response to Abraham becomes is "becomes what?" However, based on the discussion I am means Jesus claimed to be before Abraham who is dead. As Catrin H. Williams states [emphasis added]:
Jesus is "testifying to his precedence over Abraham...resulting from a deliberate distinction established between the patriarch who came into existence at a particular moment in history (γενέσθαι: cf. 1:3, 6, 10, 14) and the absolute form of being claimed by Jesus (εἰμι: cf. 1-14). Although the statement can be defined as Jesus' claim to timeless existence, more seems to be implied by ἐγώ εἰμι, particularly if elements binding these climatic words to the declarations in vv. 24, 28 can be identified.3
Regardless, of what event Jesus is referring to, "I am" in relation to Abraham's present condition (he is dead), means Jesus is placing Himself in existence before that change. More importantly, Jesus does not say was He says am. While this could imply a timeless existence, it is a claim of continual existence from before Abraham to the present. That is, Jesus is claiming to have continually existed from since before Abraham becomes (something) until the present. Clearly no human can truthfully make this claim about themself.
Before what?
If the before condition is Abraham's birth, then Jesus states He was and still is, in existence before Abraham was born. This could be taken as a claim to be God. Yet, Abraham did experience a change after his birth:
1 When Abram was ninety-nine years old the LORD appeared to Abram and said to him, “I am God Almighty; walk before me, and be blameless, 2 that I may make my covenant between me and you, and may multiply you greatly.” 3 Then Abram fell on his face. And God said to him, 4 “Behold, my covenant is with you, and you shall be the father of a multitude of nations. 5 No longer shall your name be called Abram, but your name shall be Abraham, for I have made you the father of a multitude of nations. (Genesis 17)
God Almighty, El Shaddai, made a covenant with Abram and gave him a new name, Abraham. In other words, Abram γενέσθαι Abraham. Using this event Jesus meant "before Abram became Abraham, I am."
The LXX translates the beginning of this passage somewhat differently:
Now Abram came to be ninety-nine years
of age, and the Lord appeared to Abram
and said to him, “I am your God; be well pleasing
before me, and become blameless
(LXX-Genesis 17:1 NETS)
ἐγένετο δὲ Αβραμ ἐτῶν ἐνενήκοντα ἐννέα καὶ ὤφθη κύριος τῷ Αβραμ καὶ εἶπεν αὐτῷ ἐγώ εἰμι ὁ θεός σου εὐαρέστει ἐναντίον ἐμοῦ καὶ γίνου ἄμεμπτος
The LXX changes "God Almighty" to "your God" who identifies Himself as ἐγώ εἰμι ὁ θεός σου, "I am your God."
Conclusion - Part 2
The words ἐγώ εἰμι by themselves are not anything more then I am... It is the context which adds additional meaning. For example, ἐγώ εἰμι spoken by the man who had been born blind mean, I am the one who was born blind. There is no mystery to what the man is saying.
Obviously, before Abraham γενέσθαι ἐγώ εἰμι raises a question as to what exactly Jesus is saying about Himself. Those who heard Jesus and were aware Abram becomes Abraham would easily connect what Jesus is saying to the event in Genesis. Any Jew making this connection would easily connect Jesus' claim to continual existence using ἐγώ εἰμι relative to Abram becoming Abraham as a claim to be Abram's God, which in the Hebrew was יהוה אל־אברם. For this blasphemous claim Jesus would need to be stoned:
So they picked up stones to throw at him, but Jesus hid himself and went out of the temple. (John 8:59)
1. The extent or limit of the "future" is not specifically given. Rather, the entire Gospel speaks to what John believes is, or might be future limits. For example, the reader would rightly conclude becoming children of God is permanent and so, eternal. But becoming Jesus' disciple is not necessarily permanent. Likewise the permanence of receiving the Holy Spirit is a matter of belief. Regardless of the permanence, John's use of γενέσθαι always includes the element of the future.
2. According to Jewish history, Abraham was born 1,948 years after creation and died at the age of 175 (year 2,123). Abraham had been dead for almost 1,800 years.
3. Catrin H. Williams, I Am He: The Interpretation of A̓nî Hû in Jewish and Early Christian Literature, Mohr Siebeck, 2000, p. 276