It became a serpent. The word here used for "serpent," nakhash, is a generic word applicable to any species of snake. We cannot assume that the cobra is the serpent meant, though no doubt Moses, when he fled from before it, believed it to be a venomous serpent.
Various reasons for God's choice of this particular sign have been given. Perhaps the best is, that a trick of the kind was known to the Egyptian conjurors, who would be tempted to exhibit it in order to discredit Moses, and would then be discredited themselves by his stick swallowing theirs.
(See Exodus 7:10-12.) It is fanciful to suppose a reference either to the serpent of Genesis 3:1-24.