What was Sir John Mandeville's source regarding Beersheba and its chronology?

To support the answer by @Joshua, Wikipedia says:

Beersheba is mainly dealt with in the Hebrew Bible in connection with the Patriarchs Abraham and Isaac, who both dig a well and close peace treaties with King Abimelech of Gerar at the site. Hence it receives its name twice, first after Abraham's dealings with Abimelech (Genesis 21:22-34), and again from Isaac who closes his own covenant with Abimelech of Gerar and whose servants also dig a well there (Genesis 26:23-33). The place is thus connected to two of the three Wife–sister narratives in the Book of Genesis.

In particular, the Genesis 21:31 literally says:

Therefore that place was called Beer-sheba;

This should be read more correctly as "Therefore this place is called the well of covenant " (Mistranslating sheva of Beer Sheva as seven rather than as covenant is rather common.)

Further remarks:

• Whether Abraham is a real or a mythical character is of little importance here, since the Hebrew bible was written well after the events. On linguistic evidence, its earlier chapters (including Genesis) date back to the first millennium B.C.
• It is difficult to attest whether the place mentioned in the bible is the same as the one called Beer Sheva today. The ancient site of Tel Sheva is located beyond the city line, whereas the site called "Abraham's well" in the old city is certainly a much later creation.

I read in Genesis of the naming of the place of Beersheba, where it mentions not a town but only the wells Abraham had dug. We can suggest a nearby town is present because Abimelech does not appear to be a nomad. (Incidentally we do not and cannot know if the wells there today are the same wells or not.)

But by the time of the judge Sameul it is recorded that there is already a town there and the sons of Sameul are already sitting as judges there and David has not yet sat upon the throne. Thus we may find that Sir John Mandeville is simply wrong. Nevertheless take note we do not know for certain if it's the same place as Abraham had wells dug but rather that the the people of the time of the Kings of Judah believed it was the same place. No matter what date you set for the writing of the books of Judges and Sameul you have to accept the fact there is a continuity of memory good enough to fix place names from the time of the conquest of Canaan through the exile to Babylon and the return thereof. Name-location pairs were only lost when the Romans had quite enough and kicked the Jews out for a very long time.

In answer to the other part of the question, I am aware of no source that puts the name Beersheba before Abraham and I would expect linguistic difficulty in doing so. It's good enough to either say "not named such before Abraham" or "translated into Hebrew instead of transliterated as is their typical practice".

The archeological evidence is the town at the modern-day sight of Beersheba goes back very far, to the point where any reasonable date would be before the conquest of Canaan by the Israelites.

We do not know John Mandeville's sources and this is mystery what legends he drew from, but they can't be true.

This can't really be reconciled historically, because Abraham was written as a mythological figure, not a historic one.

The Abraham story cannot be definitively related to any specific time, and it is widely agreed that the patriarchal age, along with The Exodus and the period of the judges, is a late literary construct that does not relate to any period in actual history.

As for Mandeville itself, we don't really have anything to go on other than that author's writings in that one work, so if the logic isn't there, we don't know.

We can say more generally that this was before the full development of the Scientific Method, and Mandeville's author had the sensibilities of a storyteller, not a Historian. So its a fair bet that if they made an assertion not otherwise supported by facts, it was because they thought that reading made for the best story.