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The title "The Father of All" is thought by some to be borrowed from the god Odin in Norse mythology, though the New Testament also refers to God as the "one God and Father of all". Tolkien, as a Catholic and a scholar of northern European mythology, was probably influenced by both sources. (The name itself is probably based on the Old Norse Alfóðr, a name for Odin, however.) As Tolkien was highly educated in Finnish mythology, it would be no surprise if the name of Ilúvatar were influenced by Ilmatar, Maid of Air, one of the primal spirits of creation.
It is to be noted that in earlier works of the legendarium the name Ilúvatar meant "Sky-father" since the element il- refers also to the sky (cf. Ilmen), but this etymology was dropped in favour of the newer meaning in later revisions. Ilúvatar was also the only name of God used in earlier versions — the name Eru first appeared in the Annals of Aman, published in Morgoth's Ring, the tenth volume of The History of Middle-earth.