First of all, as Denis noted, the Semitic cultures were often inter-mixed so some Jewish ancestry is not entirely out of question.
Now, Let's see the case of House of Saud. House of Saud comes from Najd region of Arabian peninsula. As per Nejedi Chroniclers, The Saudis are descended from the larger Mrudah clan. That much is at least for sure. Before that, opinions vary from tradition to tradition. The Nejedi Chroniclers claim Mrudah were themselves descended from Banu-Hanifa tribe but others dispute that and claim that their origins are in the Anazzah tribal confederation.
in Saudi Arabia: The Challenge of Succession
Be it as it may, Both Banu-Hanifa and Anazzah are themselves part of the Rabi'ah Branch of Clans.
Now going further back, the Rabi'ahs are part of the Northern/Adnanite Arabs who according to them were descended from Adnan, a descendent of Ishmael. Since Ishmael and Isaac were brothers and Jews consider themselves to be descended from Jacob, Son of Isaac, that would make them kins. It should also be noted that Prophet Muhammad's own tribe Quresh is also Adnanite.
But that of course is with making a lot of assumptions, which are the existence of the Biblical Figures and accuracy of the Arab Genealogies.
IRL however, if Genetic studies were to be made, there would be a lot of similarities between all the Semites. For example take the Canaanites, Bible said they were wiped out but DNA studies showed that Lebanese Arabs have their DNA. And if Dr. Harry Ostrer of Human Genetics Program at New York University School of Medicine can be believed, Palestinians, Israelis, Lebanese, Syrians share a common lineage, a belief shared by many others although Ostrer is often criticized for his research methods and his hypothesis on Khazars among his peers. Note that Anazzah used to live in Levant before some branches migrated to Arabia proper. In Addition, Arabic and Hebrew languages are both Semitic.
So TL;DR, Yes it is very likely that the Saudis share a common ancestor with the Jews but so do most of the other Semitic people, at least Hypothetically. As Ted noted in his comment, if we go back far enough, we are all related.