(I'm assuming that you mean an aristocratic lady here, because of the contrast between "lady" and "common man" and your assumption that his title would change. But it needs to be noted that "noble" = "aristocratic" = "has a title of some kind"; I'm ignoring your "non-noble" because it doesn't seem to make sense in this situation. If I've misjudged what situation you're describing, I'm sorry.)

Nope! In the western patriarchal system, a woman took her status from her husband - not the other way around. If she were born to a duke, marquess, or earl, she would have had the courtesy title "Lady Firstname Lastname" from birth; marrying a man without a title (or with only a courtesy title himself) would have made her "Lady Firstname Marriedlastname". If she had her title through having previously been married to a peer and being widowed, she could keep her title as a courtesy but wouldn't pass anything on to her non-titled husband.

"The Honourable Mr./Miss Lastname" was/is a courtesy title by the younger sons of earls, viscounts, and barons and the daughters of viscounts and barons. If the Honourable Miss Lastname married a man without a title, she would become the Honourable Mrs. Marriedlastname.

You might be interested in some of the answers I've written in the past on Victorian etiquette:

Would it be appropriate for the eldest son of a viscount and the eldest daughter of a baron to be wed? And if so, what would their new titles be, and the titles of their children?

In the Victorian period, how did people address each other?