Boeing Co.’s 737 MAX plane could return to service this summer, yet convincing passengers the plane is safe will be one of the aviation industry’s toughest consumer-relations challenges in decades.

The aircraft has been grounded world-wide since March after two MAX jets crashed within five months of each other. The crashes, and what some carriers and pilots have described as Boeing’s lack of transparency in their aftermath, have undermined confidence in the plane maker.

Industry...