A 'commuting wife' is, as the name suggests, a form of marriage in which the wife commutes to the husband's place.
Literal
Commuting-wife [as-for-とは] name [gen-の] as, wife [subject-が] husband [gen-の] place [to-に] commute marriage-form [gen-の] thing is.
Another definition. 通い妻 ('commuting wife') is a specific Japanese term for a non-cohabiting marriage arrangement where the wife visits the husband's home rather than sharing it. 名前の通り ('as the name indicates / true to the name') is a fixed phrase. 夫の元 ('husband's place/side') uses 元 ('source, origin, base') in its 'where someone is' sense — archaic-flavored vocabulary. ~の事である is the formal definitional copula.