It is really a question of believing what is written in Genesis 21:17-21… plus accepting the real physical evidence of the well of Zam-Zam in Makkah… plus the historical acounts of the actual descendants of Ishmael (pbuh) regarding the location of where he lived.
Is there to be found an equally plausible alternative theory of where Biblical Paran is?
There is no historical or biblical source that associates the well Hagar saw with he well in Mecca. It is impossible to get the exact number and places of the wells that existed in those days.
Paran cannot be Mecca because Mecca was not considered a place in the wilderness whereas Paran was the name of an isolated place in the wilderness. The Koran depicts Mecca as an uncultivable valley, but never refers to it as wilderness. Hagar was directed to a a well in the wilderness of Paran for sure. When it was almost impossible for her to find some water in the wilderness, God intervened and helped her.
Let’s read the narrative:
Genesis 21:14-21
And
she departed, and wandered in the wilderness of Bersabee. And when the water in the bottle was spent, she cast the boy under one of the trees that were there. And she went her way, and sat overagainst him a great way off, as far as a bow can carry, for she said: I will not see the boy die: and sitting overagainst, she lifted up her voice and wept. And God heard the voice of the boy: and an angel of God called to Agar from heaven, saying: What art thou doing, Agar? fear not; for God hath heard the voice of the boy, from the place wherein he is. Arise, take up the boy, and hold him by the hand, for I will make him a great nation. And God opened her eyes: and she saw a well of water, and went and filled the bottle, and gave the boy to drink. And God was with him: and
he grew, and dwelt in the wilderness, and became a young man, an archer. And
he dwelt in the wilderness of Paran, and his mother took a wife for him out of
the land of Egypt.
Ishmael did not live in a city, but in the wilderness of Paran. His mother took a wife for him from Egypt, which is far off Mecca.