The text of Gilgamesh does not tell us why Utnapishtim is chosen. We are not told that he is a better person or more worthy than anyone else. The only clue is that he was able to hear a secret message spoken to a 'reed wall' by the god Ea. There is also a hint that he was a devotee of Ea, based on the expression 'I will go down to the Apsu [the cosmic realm of the subterranean waters] to live with my Lord, Ea'.
In the Epic of Atrahasis, on which the flood story in Gilgamesh is based, Atrahasis/Utnapishtim is described thus:
Now there was one Atrahasis
Whose ear was open to his god Enki.
He would speak with his god
And his god would speak with him.
So it seems that Atrahasis was someone who worshiped and could communicate with Ea (also known as Enki). In this epic, Ea, who is renowned for his wisdom, helps Atrahasis rescue humanity from a number of plagues and punishments sent by the chief god Enlil. The flood is merely the final and ultimate punishment in this series. It should be noted that in Mesopotamian tradition, Atrahasis ("extra-wise") is the flood hero's name prior to the granting of immortality, after which he is known as Utnapishtim ("he found life" or "he saw life").
In Sumerian, this same flood hero is known as Ziusudra and he is identified as a king, priest and seer. In one of the Sumerian king lists he is listed as king of Shuruppak before the flood, and he is also associated with this city in Gilgamesh.
Part of the answer to your question, then, is that Utnapishtim was chosen because of his relationship to the God Ea. Equally if not more important, however, is the purpose for which Utnapishtim wa chosen. As mentioned above, we are not told that Utnapishtim was more righteous or worthy than his contemporaries. Rather he is the instrument thorugh which Ea saves human and animal life from complete destruction in the flood.
It should be noted that Utnapishtim was chosen by Ea alone - not 'the gods' as a group. In fact, it was the council of the Gods that agreed to exterminate humanity and Ea deliberately thwarts this decision. In the final showdown between Ea and the chief god Enlil, the other gods take Ea's side and Enlil is accused of sending a disproportionate and unjust punishment. Enlil seems to accept this criticism by granting immortality to the flood hero and his wife as a reward for saving living beings from extinction. In the larger context of the epic this constitutes a subtle rebuke to Gilgamesh, who has exhausted himself seeking personal immortality while neglecting the well being and care of his people.