Neh. 2:13 from Young's Literal Translation (YLT) reads:
"And I go out through the gate of the valley by night, and unto the front of the fountain of the dragon, and unto the gate of the dunghill, and I am measuring about the walls of Jerusalem, that are broken down, and its gates consumed with fire."
It is not certain where this well was, nor why it was so named.
Gill's Exposition of the Entire Bible has:
"even before the dragon well; so called from its winding about, just as a crooked winding river is called serpentine; though some think here stood an image of a dragon, either in wood, or stone, or brass, out of the mouth of which the water flowed from the well; and others, that since the desolations of Jerusalem, serpents or dragons had their abode here: " Souce: here
While the Cambridge Bible for Schools and Colleges states:
"even before the dragon well] R.V. even towards the dragon’s well. The LXX. misreading the less common word in the Hebrew for ‘dragon,’ renders καὶ πρὸς στὸμα πηγῆς τῶν συκῶν. The name is doubtless connected with some sanctuary at this fountain in prehistoric times, when ‘living water’ was associated with the worship of a deity often represented by a ‘dragon.’ Prof. Robertson Smith (Religion of the Semites, pp. 156, 157) calls attention to ‘the connection of jinns in the form of dragons or serpents with sacred or healing springs’ … ‘The river of Coele-Syria, the Orontes, was carved out, according to local tradition, by a great dragon, which disappeared in the earth at its source.’ The explanation that the well was so called ‘because some curious large watersnake or crocodile was kept in it in Nehemiah’s time’ may be disregarded as fanciful and improbable.
The identification of the well is uncertain. By some it has been identified with ‘En-Rogel,’ near ‘the serpent’s stone,’ the stone of Zoheleth (1 Kings 1:9). But see Nehemiah 2:15. By others it has been identified with ‘the fountain of Gihon’ (1 Kings 1:33). Neither of these suggestions suits the present verse, from which we gather that ‘the dragon fountain stood on the W. or S.W. wall of the city.’ If it was a spring in the Tyropoeon Valley, it has long since been choked up. ‘The rock-hewn conduit which has been found running along the bed of the Tyropoeon Valley’ (Sir Chas. Wilson’s Jerusalem, p. 113, 1889) may very well have conducted the water from such a spring. The ‘serpent,’ or Mamilla Pool, lay at the N. end of the modern Hinnom Valley (Joseph. Bell. Jud. v. 3. 2)." Source: Ibid.
I did find a reference source (for purchase) from the Palestine Exploration Quarterly, Vol. 30, 1898, Issue 4 by Dr. Conrad Schick who seemed to think that most people today have incorrectly associated this well with the Birket Mamilla which is too far west. As Nehemia was traveling south Schick suggested it was actually the lower pool at Birket es Sultan and was located in a rock cave where a seasonal spring would come up.
You can read part of this info here.