There's a number of really cool day trips near the Benson area:
1) Fairbank, AZ - In between Benson and Tombstone there's a neat little historic ghost town called Fairbank that's just a short detour down HWY 82. There's some neat old buildings and a little schoolhouse museum/bookstore that's open on the weekends. It's located in the San Pedro Riparian National Conservation Area. There's a great easy nature trail loop to hike as well.
2) Ghost Town Trail - A nice scenic loop that has a terminus in Tombstone. From Tombstone it runs East/North East through Gleeson, Courtland, and Pearce. There's a great little museum in the old jail house in Gleeson which is owned, curated, and maintained by Joe Bono who grew up in the town before the mining boom ended. Further down the trail from Gleeson you'll run across the ruins of the old Courtland jail house, and eventually you'll arrive in Pearce, which has a few fantastically restored buildings including an old U.S. Post Office.
3) Bisbee, AZ & Lowel, AZ - Bisbee is a wonderful little town with a ton of great gift shops and galleries along the 'downtown' Main St. There's a few great museums and even a mine tour, the Queen Mine Tour, that will take you in an actual mine train deep into the mountain side for an unforgettable tour. Just South of Bisbee is a one road "town" called Lowel, AZ that will take you through a time warp back to the middle of the 20th century. The one block long "town" serves as a living museum filled with interesting old cars and signs.
4) Slaughter Ranch / San Bernardino National Wildlife Refuge - About 10 minutes East of Douglas, AZ along Geronimo Trail and the US / Mexico border is an absolute hidden gem of the state, in my opinion. It's probably my favorite place to visit along the Arizona Borderlands. Slaughter Ranch is a wonderfully restored and maintained ranch from around the turn of the 20th century. There's incredible history there, and a great self-guided walking tour that explains much of the important history of the area. There you can also walk around the ruins of an old U.S. Army fort where soldiers were able to observe some of the movements of Pancho Villa and his army during the Mexican Revolution. The ranch is a desert oasis with something like 20+ natural artesian wells on the property. It's a fantastic spot for a picnic, there's also charcoal grills and tables available for use by anybody. The caretakers are wonderful people who also take care of a number of animals on the property, including a number of dogs that have been taken out of traumatic situations. Because of this, NO DOGS ARE ALLOWED, so plan ahead of time if you have a pet with you. They ask for a $5 donation for each adult visiting - it's well worth the price.
5) Chiricahua Mountains / Chiricahua National Monument - The Chiricahua Mountains is one of the more interesting mountain ranges i've ever seen. There's really crazy rock spires and deep narrow canyons. Absolutely magical place. There's a ton of trails all around the mountains, and then there's the National Monument which has a paved scenic drive, great hiking, lots of scenic picnic spots, a couple museums and a gift shop