I do not expect any other pronunciation than /miɲɔt/ by a French person. It is close to (UK) English "min¹ yacht", the middle "gn" is pronounced like the Spanish ñ, close to "canyon", the name rhymes with "pot".

Mignotte is not just a proper name but also a noun/adjective in French. Despite being relatively outdated and replaced by mignonne, it is still sometimes used and always pronounced /miɲɔt/:

TLFi (archive)

MIGNOT, -OTTE, subst.
Synon. vieilli ou région. de mignon (dont il est la forme anc.). Viens ça, mignot, dit le poète, à qui sa chaire en vieux chêne donnait parfois des velléités de «vieil langaige» (A. Daudet, Jack, t.1, 1876, p. 282). Ô ma belle mignotte, mon rêve adoré (Huysmans, Le Drageoir aux épices, p.22 ds Cressot, Phrase et vocab. Huysmans, 1938, p. 511).
…
Prononc.: [miɳ ο], fém. [-ɔt].

Note: The TLFi seems to confuse /ɳ/ and /ɲ/, the former being unused in French (of France).

This last name origin is reported to be Burgundy (archive):

D’où vient ce nom ?
Mignot
Nom très courant dans toute la France, notamment en Normandie et en Auvergne. C’est un adjectif qui signifie en ancien français joli, gracieux, aimable. Matronyme : Mignotte (Bourgogne).

¹ _{The vowel is closer to the one in mean, but is short}