Mollycoddle is both a verb and a noun. It appears that the noun came first, as a descriptor for a weak, sissified man. The construction probably derived from molly, which is an Irish diminuitive of Mary, which came to mean any woman, and from coddle, in the sense of treating gently or pampering. Coddling, in 19th Century English, meant to nurse, to protect, to treat with exceeding gentleness, as one would an invalid or ill patient. The modern sense of the verb is to treat with excessive indulgence.

When a recipe for Dublin Coddle warns the cook not to let the broth boil, there is a sense in which the potatoes in it are being coddled -- pampered, treated gently.

That, it strikes me, is the connection ... but I could be wholly wrong, and not for the first time.