David was the sixth and youngest son of of Malcolm III and St Margaret, who was an English princess of the House of Wessex. The others, in order, were Edward, Edmund, Edgar, Ethelred and Alexander: four emphatically Anglo-Saxon names, which had also been borne by members of Margaret's immediate family, and one classical one.

There has been a good deal of speculation about the names that Malcolm and Margaret chose for their children. It's certainly very notable that the three sons the king had had with his first wife, the daughter of Thorfinn the Mighty, Earl of Orkney, all had Scottish names - Duncan, Donald and Malcolm – but the Saxon names bestowed on the children of the second marriage don't necessarily show that Margaret was the dominant force in the union; it's at least as likely that the couple hoped that William I's new Norman regime to the south would prove to be unstable and offer them an opportunity to place one of their sons on the English throne; it thus made sense to give the eldest of them names that stressed their half-Saxon parentage and which would be acceptable to the English.

As for Alexander and David - it's usually speculated that "Alexander" – the name of history's greatest general – was a nod to Malcolm III's status as a warrior king, and that "David" was chosen because Margaret, by then in her late 30s, expected him to be her last and hence youngest child – just as the Biblical David was the youngest of eight sons. (Actually she went on to have one more child, a daughter.)

There would have been very few Jews indeed in the Scotland of the time (our sources suggest that the first significant immigration of Jews into Britain did not take place until after the Conquest in 1066), so it is more realistic to suppose that contemporaries thought of "David" as a Biblical name rather than as an explicitly Jewish one. More pertinently, it's probable that Malcolm and Margaret felt less constrained to choose the contemporary equivalent of "politically correct" names for their last two sons since it must have seemed highly unlikely either would succeed to a throne. It is one of history's small ironies that both of them did.

From the point of view of your debate, I'd suggest that it's no more plausible to argue that David I of Scotland got his name because his parents thought they were descended from the kings of the Old Testament than it is to argue that his brother Alexander got his because they thought the kings of England and Scotland were related to the Macedonian dynasty of Philip II. Also, if Malcolm and Margaret had been desperate to mark their Jewish heritage, he'd spurned eight prior opportunities to do do, and she five.

Sources

CNL Brooke, Churches and Churchmen in Medieval Europe

Richard Oram, David I: The King Who Made Scotland

Because it's a biblical name and because Christianity in the British isles dates back to very early. Probably at least the 2nd century for 'England', the 5th century for Ireland, and Irish missions into the Scottish north/west in the 6th century. Various biblical names spread and were adopted by Christians for use as common names, and it's not remarkable at all.

There's nothing really that odd about David being the King's name, no more than the 3 Alexanders also in the Dunkeld line, or the 3 Constantines listed as Pictish (and later Scottish) Kings.

Certainly no evidence for British Israelism in the slightest.