As noted in other answers, the Greek ἄνωθεν anōthen means either again or above. Since John records Jesus using the word twice, each meaning can be applied:
See the Kingdom of God - born again
Enter the Kingdom of God - born from above
The response to Nicodemus' question about a man returning to his mother's womb, is "No. For a man to be born again (a second time), requires an act of (re)creation; a man must be born from above (from God)." This sequence follows the central theme of the Prologue:
But to all who did receive him, who believed in his name, he gave the right to become children of God, who were born, not of blood nor of the will of the flesh nor of the will of man, but of God. (John 1:12-13 ) [ESV throughout]
A person has the authority to become a child of God: they can be born again. This authority is not from earthly rights or privileges. It is from the will of God: they are born from above.
The passage, obviously is understood solely from the Christian perspective. Yet the Gospel indicates a different type of "rebirth" is also possible:
He said to them, “You are from below; I am from above (ἄνω). You are of this world; I am not of this world. (John 8:23)
First, Jesus is from above ἄνω. The word anōthen is from ἄνω which means from on high or of time, formerly [anō]. As with anōthen, both meanings are intended. Jesus is anō; He is from above and from before. The others are from below. The word here is κάτω which means below or beneath either of place or temporal succession [katō]. As with anōthen and anō both meanings can be applied. The others have been "reborn" from below:
You are of your father the devil, and your will is to do your father's desires... (John 8:44)
A person will see the Kingdom of God when they have been reborn, from above or from below:
And if I cast out demons by Beelzebul, by whom do your sons cast them out? Therefore they will be your judges. But if it is by the Spirit of God that I cast out demons, then the kingdom of God has come upon you. (Matthew 12:27-28)
Seeing the kingdom of God does not mean entrance, because those born from below also see God's Kingdom, but their perspective is negative. In other words, a person who is born from below sees the kingdom of God as something to oppose.
Therefore the complete teaching about rebirth is:
Jesus answered him, “Truly, truly, I say to you, unless one is born again [either from above or from below] he cannot see the kingdom of God.” (3:3)
Jesus answered, “Truly, truly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God. That which is born of the flesh is flesh, and that which is born of the Spirit is spirit. Do not marvel that I said to you, ‘You (plural) must be born from above.’ The wind blows where it wishes, and you hear its sound, but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit.” (3:5-8)
Finally, unless one is born of water and of Spirit, they remain as one "born from below" and the second use of anōthen would be "again" not "above".