Lymington is a nice small town for strolling around during the summer. They do have many shops and restaurants around for such a small place.
We went to the Stanwell House Hotel as a pretty much random choice.
I had the monkfish in bean broth which was very tasty and surprisingly filling. Angela had the tempura style red mullet but that was a bit greasy.
Not exactly somewhere cheap to pop in for a quick bite to eat, but very pleasant. If it is a nice day you can sit and dine outside in the garden at the back.