As noted in D.A. Hosek's answer, Malory’s Le Morte d'Arthur was the primary source for White. In the Morte d'Arthur there is no mention of Merlin having an owl, or indeed any pet/familiar at all. Even physical descriptions of the wizard are very brief. Instead the character of Merlin in the Once and Future King, was not based on Arthurian legend, but on another source - White himself. As White's biographer, Sylvia Warner, noted, the character and eccentricities of Merlin are clear images of White. He had recently been a schoolteacher at Stowe school, and took the opportunity to make Merlin what he considered to be an ideal schoolteacher, operating in an ideal educational environment:
he gave himself an ideal old age, free from care and the contradiction
of circumstances, practising an enlightened system of education on a
chosen pupil, embellished with an enchanter's hat, omniscient,
unconstrainable and with a sink where the crockery washed itself up.
As Merlyn, White had the time of his life: the brief dazzle of being
head of the English Department at Stowe was a farthing candle to it.
Apart from furnishing Merlin with many of his most prized possessions - his beard, his knitting, his opinions - he also gave him possession of his pet owl. White kept quite a menagerie of animals, and at the time it consisted of his beloved red setter and some birds:
He went on keeping Cully [a goshawk] and two sparrowhawks and presently added an
owl, whom he called Archimedes
[A hawk called Cully is also familiar to readers of The Sword in the Stone.]
As to why he chose the name Archimedes, Warner does not give any mention, nor is it mentioned in any of White's published letters (The White - Garnett Letters, and Letters to a friend : the correspondence between T. H. White and L. J. Potts). But I think the association of owls with wisdom and Athena, and thence to Greek culture, makes the choice of the name of a Greek philosopher fairly natural.
