Hartlepool isn't a foodie destination. The marina has the predictable chain restaurants — Italian, steakhouse, the inevitable Wetherspoons — and they do the job for a quick lunch between museums.
For something more local, try the fish-and-chip shops on the Headland or in Seaton Carew; Hartlepool's chippies are taken seriously locally and the fish comes in fresh off the small boats. The Causeway pub on the Headland does decent pies and ales.