EVERY TOURIST in Arizona knows about Tombstone ("The Town Too Tough To Die"). On the other hand, Bisbee is not exactly a household word, pardner, even though the sheriff of Cochise County still rides the range out there.

Bisbee (pop. 8,328), 25 miles south of Tombstone and six miles north of Mexico, rates of brief mention in the Mobil Travel Guide's latest edition. But the former turn-of-the-century boom town has a passel of problems that a modest influx of tourists hasn't solved.

As with many Arizona boom towns - Total Wreck, Goldfield, and Tombstone among them - Bisbee's bubble burst. In 1974 the "company store," the Phelps Dodge Corporation, elft town when ore became scare there and copper prices didn't rise.

Bisbee was known as the "Queen of the Copper Belt," with production raching the $2 billion mark, but the town suddenly lost 1,800 working men and the population went from 12,000 to 7,600. It hasn't recovered yet. It is $500,000 in debt, 29 city employees have been laid off, there many soon be no police force, and there has been talk of a mayoral recall effort.

To make things more interesting, Bisbee is composed of a not-always-harmonious population of Mexicans, former mining officials, Army personnel from nearby Fort Huachuca, and what some old-time residents scornfully (yet sometimes affectionately) call "the hippies."

They came seeking a tranquil place far from the turmoil of the '60s and the complexities of the big city. Word spread in the Southwest of an abandoned mining town nestled in the Mule Mountains where the weather was perfect year 'round and homes could be bought for as little as $500.

Old, hillside mining shacks were turned into quaint, colorful homes that frequently attract professional and amateur photographers. Bisbee is said to resemble early San Francisco - judge De Witt Bisbee, who in 1880 financed Bisbeehs first copper-mining smelter.

New businesses, including art galleries, jewelry shops processing "Bisbee Blue" turquoise, book stores and silver shops appeared along with restaurants where the traditional fare of beer and whiskey was replaced by yogurt and bean sprouts. A theater group recently performed "Inching Through The Everglades," an old movie house was brought back to life, and cable TV has been introduced.

The history of Bisbee is still evident from its old buildings and the once-famous Brewery Gulch District, which in its day sported 40 bars, each with a liberal supply of ladies of the evening. One saloon featured a bear that would get drunk with indulgent miners, then wrestle with them. As the story goes, the bear, loaded one evening and chained to a cottonwood tree, climbed on a high branch and hanged itself.

In its day, Bisbee was reputed to be the most memorable pause between New Orleans and San Francisco. "Black Jack" Pershing stayed at the opulent Copper Queen Hotel before continuing his futile search for Pancho Villa. Teddy Roosevelt was also a guest.

The Copper Queen, completed in 1902 to provide luxury accommodations for copper purchasers and mining experts, has been restored to its original elegance with the addition of a second-floor swimming pool.The hotel features a saloon and a dining room, each with an outside terrace, rocking chairs, and a view of the mountains. Musicians, including fiddle and piano players, wander through the saloon on no particular schedule, and play.

Limited mining is still done in Bisbee through a method called leaching, but nowadays only a mine tour and some bitter memories of the past call attention to Bisbee's mining history.

In 1917, 1,500 members of the IWW, the International Workers of the World, went on strike in Bisbee. Sheriff Harry Wheeler and his deputies rounded up the workers, also known as Wobblies or "I Won't Works," and put them in cattle cars. They were "deported" to Rodeo, N.M. The event has been called the largest kidnapping in U.S. history.

You won't find too much information on that event in Bisbee. It is not chronicled in the local museum. But you can find out something about the first miners by taking the tour of the Copper Queen Mine.

Underneath Bisbee's mountains lie nearly 2,000 miles of mining tunnels that took about 100 years to construct. Former miners take visitors wearing mining hats, lights, and jackets 1,800 feet into passageways dripping with water and spotted with grafitti. The temperature is 47 degrees. The guides explain that the first miners earned 15 cents an hour and worked a 10-hour day. Ore was extracted with the help of 125 burros who never left the mine unless they became sick. They were then taken above ground where they went blind from sudden exposure to the light. Some would go crazy.

While a statute in Bisbee pays tribut to the miners, there is none honoring the burro.

In its day, Bisbee had streets cars, two opera houses, a Civil War-era steam engine to haul the ore, and a brewery that flourished until the railroad brought in iced beer. Currently the town is becoming known in international circles because it is now the site for yearly bicycle races in which competitors from throughout the world meet in four grueling events. The Bisbee races have become mandatory competition for the Olympics.

Despite all this, the town is groping for a way to get out of debt and return to peacefulness, the quality so many sought when they came to Bisbee.

Bisbee Mayor Chuck Eads recently endorsed a $4.50 garbage tax to get the pass the city council. An investor from Tucson who offered to buy the town received a cool reception from many of the city's officials.

Fears abound that Bisbee, like Tombstone, may become merely a laughable curiosity. Others cynically suggest that it will be ruined by wealthy investors, turned into what one former Silver Spring, Md., resident called "another Georgetown where hamburgers will cost $4."

However, William C. Epler, publisher of the Brewery Gulch Gazette and a mining publication called Pay Dirt, and who is aware of Bisbeehs current political agonies, stresses the town's future. Elper says bank deposits have increased, post office receipts are rising, utility connections are increasing and a state report shows population growth since 1970. Epler adds that Bisbee benefits from being a "bedroom community" for Fort Huachuca.

Mayor Eads believes the town will blossom because of a growing number of tourists and its reputation as a retirement community.

Bisbee's narrow streets shwo signs of restoration, and everywhere saws whine and hammers pound as the rebuilders try to carve out a future. A hamburger at the Copper Queen Hotel costs $3 and there is talk that the town may soon gets its first condominium.

If you are ever in Tombstone to see the graves on Boot Hill and hang out at the OK Corral, mosey on down the road to Bisbee and find out what happened to it. CAPTION: Pictures, l and 2, Bisbee, Ariz., in 1908, top, was a bustling boom town, but today it's a town plagued with problems; By Harrison Shaffer III.