I believe that the closest end of the Grand Canyon is at least 2 1/2 hours away by car. The closest town to Lake Mead on the Nevada side is Boulder City, I think ... and that's about 40 minutes from the Las Vegas Airport. There is an 'ok' motel-type inn there called Boulder Inn (you can see it online). Are you arriving on a weekday or weekend? Weekday prices (Sun thru Thurs) are less expensive. There is a Hampton Inn on Tropicana Blvd (across the 15 Hwy), not far from the LV Airport .. very clean and serves a warm complimentary breakfast in the morning). Check online .. rates change almost daily ... you could catch it at a low price. Also, very near the airport is the mega MGM Grand, which offers very low rates if you are good about checking rates online. Also, the old Tropicana Hotel .. very inexpensive, especially weekdays. Annnd, of course, there's Hooters Hotel very near the airport ..but, LOL, I don't think I'd go there with the Boy Scouts.
Are you definite on the Grand Canyon? If not, there is a great hiking area in Red Rock Canyon (civilians and military use it for hiking, camping, etc... check it online) .. and near by is the brand new Red Rock Canyon Casino Resort, right outside the community of Summerlin. There are several forum members here from the Las Vegas area .. hopefully they'll look in and post. I live in California, so I only have limited knowledge about LV. Good luck!