After some inquiries and Internet trawling, we can answer part of your question, although sadly not the whole thing, at least not to our satisfaction. There's also some confusion amongst the available sources, which seems to stem from the property having been alternately deemed to be located at 2440 or 2460 Las Vegas Blvd. South -- a discrepancy that we're assuming stems from the store's size and the fact that the site now incorporates both mailing addresses.
Our historical research took us back to 1971, when a casino called The Big Wheel, whose big round marquee has been retained ever since, albeit with successive customizations, opened at #2440 Las Vegas Blvd. South, at the northwest corner of Sahara. Apparently, it operated under that name for about two years, advertizing 25-cent craps, 50-cent blackjack, "free beer," and "free champagne," whereupon it was renamed simply The Wheel, which debuted in 1974 but was closed within three months.
The property's next incarnation, which retained the same big round signage, albeit drastically modified, was as the Center Fold, a casino with a live topless cabaret that operated under that name from 1975 until '77, when it became the Jolly Trolley (see QoD archives for more on that story). The Jolly Trolley, which apparently featured male strippers as well as female (check out the signage in one of the images reproduced below, which proclaims "Macho Male S...[trippers?], Ladies Only") closed in 1978, then reopened in 1981 for less than a year, according to sources we've read, before closing for good.
This is where definite confusion creeps in, since all sources we've come across say that Bonanza Gift Shop opened in 1980, "next door" to the Jolly Trolley, which was indeed still in business when the store debuted. What we're assuming is that the Bonanza first opened at #2440, the previous site of Honest John's casino (see photo), and then expanded next door into #2460 when the Jolly Trolley closed.
Calls and emails to the Bonanza Gift Shop which, with its 40,000 square feet of shopping space, in true Vegas tradition bills itself as "the biggest gift shop in the world" (no proof is offered and a store in Missouri boasts the same claim), yielded no response, and our online sleuthing bizarrely yielded no further insights into the story of one of Las Vegas' more famous landmarks, other than the fact that the current owner is Ms. Julia Lynn Morris, who's occasionally called on by the local press to discuss current trends in the souvenir market. But that's about all we've got, so if anyone else knows more about this elusive site's current occupant and its history, please drop us a line.
Oh, one more thing: Should you find yourself making a pilgrimage to this must-see Las Vegas attraction, where you can pick up anything from a "Howdy Pod'ner" comedy cactus antenna for your car, to a classic dice clock, to a "Don't leave me in the car. It's hot!" baby onesie, not to mention playing cards, dice, poker chips, ashtrays, t-shirts, signs, and every other piece of Vegas kitsch imaginable (plus a few we know you couldn't possibly have thought of...), know that you'll be in esteemed company. According to the official website, past celebrity customers have included Michael Jackson, Angelina Jolie, Ringo Starr, Nicolas Cage, Cindy Crawford, Geena Davis, John Kerry, Penn & Teller, Debbie Reynolds, Condoleezza Rice, Arnold Schwarzenegger, Rod Stewart, Carrot Top, and Rip Taylor. (What this unlikely group could possibly have in common with each other could be a good brain teaser for your next trivia quiz!)