Given its superficial similarity and that all examples of its use on Google Books appear to be from Sicilian authors, it seems that Arco di Noè clearly derives from the Sicilian arcu di Nuè.
Researching a bit further, it seems another Sicilian term for rainbow is arcu di Nuvè.
Contrary to the claimed etymology on wiktionary ("arcu" + "Nuè"), the following Wikipedia pages claim that archinuè is in fact a contraction of this other term:
Archinuè (lu nomu veni a diri "Arch'i Nuè", cuntrazzioni di "Arcu di Nuvè")
Il nome deriva dalla parola siciliana Arch'i Nuè (forma contratta di Arcu di Nuè) che significa letteralmente "arco di Noè" (cioè l'arcobaleno).
So it seems that the origin of the term may be unrelated, but the modern (contracted) form may have been influenced by either the phrase L'arca di Nuè2 or the rainbow (arco) sent by God as a symbol to Noah in Genesis 9:3
- Arcu di Nuvè (Arc of Clouds?1)
Notes:
1. nuvulusu = cloudy 1 2
2. Possible backformation by analogy with L'arca di Nuè.
3. In many European languages, the term for rainbow is "Arc of [religious figure]". As per the comments above, the corrupted term name may have been influenced by Genesis 9, in which Noah is explicitly associated with a rainbow:
.יג אֶת-קַשְׁתִּי, נָתַתִּי בֶּעָנָן; וְהָיְתָה לְאוֹת בְּרִית, בֵּינִי וּבֵין הָאָרֶץ
13 I have set My bow in the cloud, and it shall be for a token of a covenant between Me and the earth.
.יד וְהָיָה, בְּעַנְנִי עָנָן עַל-הָאָרֶץ, וְנִרְאֲתָה הַקֶּשֶׁת, בֶּעָנָן
14 And it shall come to pass, when I bring clouds over the earth, and the bow is seen in the cloud,
טז וְהָיְתָה הַקֶּשֶׁת, בֶּעָנָן; וּרְאִיתִיהָ, לִזְכֹּר בְּרִית עוֹלָם, בֵּין אֱלֹהִים, וּבֵין
.כָּל-נֶפֶשׁ חַיָּה בְּכָל-בָּשָׂר אֲשֶׁר עַל-הָאָרֶץ
16 And the bow shall be in the cloud; and I will look upon it, that I may remember the everlasting covenant between God and every living creature of all flesh that is upon the earth.'
Further, Usi e costumi, credenze e pregiudizi del popolo siciliano, Volume 2, by Giuseppe Pitrè (1889) claims on p55:
VI. L’ Arcobaleno.
Arcu di Nuè; darc d’ Nuè (S. Fratello); arci ’i Diu (Naso).
Quando il mondo era per finire, Dio creò l’arcu, e disse: “ Chi si vuol salvare, salisca su quest’ arco „. Coloro che salirono, si salvarono (Montevago). (Evidentemente, qui si confonde l’arcobaleno con l’arca, e si fa salire su quello coloro che salirono sull’area nel diluvio, secondo la Scrittura).