Hi Earl –
There's no reference in the Gospels nor in Acts of Our Lord ever getting to Beersheba. Eusebius says the van broke down at Hebron and they called it a day, Jerome says someone put sand in the gas tank....
The place does figure in the OT. Abraham raised a shrine there to his God, taking the Canaanite name El Olam (Heb: ‘God of Eternity’). The place figures in the narratives of Abraham and Isaac and Jacob.
It was a place of pilgrimage, and the folktales accorded to it were subject to Yahwist and Elohist redaction. It was condemned by the prophet Amos - probably for back-sliding into a more Canaanite style of worship - and appears to have paid the price for such during the reforms under Josiah.
Thomas