Peter Lenkov knows how to make new TV shows out of old hits, and that’s a specialty in high demand at major broadcast networks these days. “Hawaii Five-0,” his update of the 1970s detective series, is in its ninth season on CBS. He also oversees “MacGyver,” a revamp of the 1980s show about a handy secret agent.

For his third resurrected series, “Magnum P.I.,” which launched this fall, Mr. Lenkov followed the same mandate. By relaunching the Hawaiian adventures of Ferrari-driving private eye Thomas Magnum, he hopes to trigger...