Named after the rue Montgallet, after a landowner or a local place name.
The City's street register traces the name to a Mr Montgallet, one of the owners of land along the street. The old lane, already on a map of 1672, was earlier the rue des Six-Chandelles and the rue du Bas-Reuilly; it appears as rue Mangallée in 1709, rue Mont-Gallet in 1760 and rue de Montgallet in 1775. Wikipédia instead links the 1709 form to a local place name.