Which of the following two cities should I visit for a one day and overnight stay, while on a north bound east coast driving trip? New London/Mystic or Providence? [I have not visited either city.]

They are totally different types of places.
Providence is a small, recently rejuvenated city, with some beautiful historic neighborhoods, and a college town feel too (Brown University and RISD supply that). Good eating and shopping there.
Mystic is all about Mystic Seaport. If you are interested in that and are going at an appropriate time (I went once in January--BIG mistake!)that will take up your time.

I agree.

Not really responding to your question, but I would opt for Newport!

Agree with all the above, including the Newport suggestion (assuming you haven't been, of course).

From Paul the writer. Thank you all, especially the Newport suggestion.

Topping.

I think if you could schedule an overnight stay in Providence on a night when a WATERFIRE was scheduled you would have one of the most wonderful nights ever! It is an event of such scale, that simply fills the night air with music & beautiful bonfire lit river.
http://www.waterfire.org/
You could tour Mystic in the day light hrs...