From Old Japanese, though the original pronunciation of the initial miso is likely unknown.
The term itself is derived from the 連用形(ren'yōkei, “stem or continuative form”) of verb 禊ぐ(misogu, “to performablutions (on someone)”, archaic).[1]
Some sources[1][2] suggest that this might be derived further from a compound of 身(mi, “body”) + 注ぎ(sosogi, “pouring over, into, onto”), or alternatively from 水(mi, “water”) + 注ぎ(sosogi, “pouring over, into, onto”). Considering that 身 had an ancient reading of ⟨mi2⟩, while 水 had an ancient reading of ⟨mi1⟩, the latter is a better potential match.
ki₁mi₁ ni yo₂ri ko₂to₂ no₂ sige₂ki₁ wo purusato₁ no₂ Asuka-no₂-kapa nimisogi₁ si ni yuku
Since thy speeches became more frequent, I’d go [back] to the old capital doing purificationin the Asuka river.
Note: Another variant has the red text replaced with 龍田超三津之濱邊爾(...Tatuta ko₁ye Mi₁tu no₂ pamabe₁ ni..., “...pass by Tatsuta and [go do purification in] the shores of Mitsu.”).