Although the Arabian peninsula doesn't play a prominent role in either the external history or internal stories of the text colloquially known today as the Arabian Nights, it's reasonable to assume that one of the most popular literary works of the pan-Arab Middle Ages was widely circulated, in some form, in medieval Arabia.

Nights consists, at its core, of a framework story in which Scheherazad tells stories, and those stories themselves (some of which feature their own internal storytellers). Like other medieval texts of this format, most famously Chaucer's Canterbury Tales, the actual number and contents of the stories in different manuscripts varies--probably one of the benefits of using such a framework in an age where book materials were scarce and every text had to be copied by hand.

Scholars generally posit a Persian (language and location) origin for the framework story in this form, although some have seen potential links to an earlier Indian/Sanskrit tradition of storytelling to keep the favor of a king. Early references to the Nights mention a Persian story of 1000 tales OR 1000 nights that has been translated into Arabic. Some of the tales, too, reflect Persian folklore.

Once the text/concept was translated from Persian into Arabic, we can trace two basic traditions of the text: one associated with Syria and one associated with Egypt. One thing is for sure: the lack of 1000 actual stories in a given copy of the text, which will be 1001 (Alf layla wa layla) by ~1150, bothered people from Muslims in the Middle Ages to European publishers in the modern era. The Syrian recension, most generally, contains fewer tales, and medieval commenters mention this with a sense of disapproval. When Galland introduced the text to Europe in the 18th century, he was indeed relying on a Syrian manuscript--but he added many tales of unknown origin (he claimed they came from oral tradition) to boost the story count. But this is only faithful to the tradition of the Nights: the Egyptian manuscript recension boosts the story count immensely.

Okay, so, to this point we have two basic versions of the text in the Middle Ages, neither of which comes out of Arabia. How can we assume it was widely known there (when manuscript evidence is understandably scarce, as it always is in the Middle Ages)? Two reasons. First, the oldest fragment of the text actually comes from 9th century Egypt, which at the time was kind of a cultural backwater. Additionally, the scrap is written on paper rather than the cheaper papyrus, reflecting a certain importance or prestige attached to the text. If the Alf Layla (it is still 1000 at this time) was already known and treasured in pre-Fatamid Egypt, it probably had wide circulation from early on.

Second, our first reference to One Thousand and One Nights is from, of all things, a Jewish source. The Cairo Geniza, an amazing and fortuitous survival of medieval documents from the (Arabic- as well as Hebrew-speaking) Jewish community in Cairo, includes a reference to a bookseller loaning out a copy of the text. The text was popular enough by the 12th century that everyday non-Muslims--not just scholars at a caliph's court--were reading it eagerly.

So even though the modern label Arabian Nights is something of a misnomer, the medieval 1001 Nights was popular enough elsewhere in the pan-Arab world to suggest it was popular in Arabia as well.