"foolish, stupid," 1530s, from Latin fatuus "foolish, insipid, silly;" which is of uncertain origin. Buck suggests originally "stricken" in the head. But de Vaan says from Proto-Italic *fatowo- "of speech," from PIE root *bha- (2) "to speak, tell, say."
[I]f we connect the fact that Fatuus is said to be an alternative name for Faunus, and that he predicted the future, and that this god is attested on an Etruscan mirror as Fatuvs in a clear oracular function (Weiss 2007b), we may venture a derivation from for 'to say' (Untermann 2000). The name of the god would then have come to be used pejoratively as 'silly'. [de Vaan]
Related: Fatuously; fatuousness.