As Rallen said, if you have to pick only one, it would be Yellowstone Park, and nearby Teton Park/Jackson Hole while you are there, also. I grew up in Wy. and there is so much to do outdoors there....fishing, hunting, backpacking, climbing, cycling, windsurfing....
Another place to see while in Wyoming is Devils Tower. An amazing geological monument. Plus it is the very first declared United States National Monument. If for nothing else you need to see it for the imposing mass that it is and to learn the history of it both geologically, and in the heritage and lore of the native americans in the area.
You can make a nice loop throughout the state and see some great stuff! Start in the south west corner, Flaming Gourge, travel north through Hoback Canyon, Jackson, Tetons, Yellowstone, Cody, then south to Thermopolis (hot springs) north to Worland and over the Big Horns to Devils Tower. And short hop to the Black Hills.
The lack of crowded cities. Cheyenne is the biggest with only ~60,000. There are only 500,000 in the whole state. There were more than that in the neighboring COUNTY where I grew up.
Old west history. Pretty much anyone you can think of, either lived here or passed through here.
The varied landscapes of Yellowstone/Tetons, Flaming Gorge, Bighorn Mountains, Black Hills, Snowy Range, as described above.
The Mint Bar in Sheridan, The Million Dollar Cowboy Bar in Jackson.
Oh, that's right. You just wanted one reason. Sorry!
a driving experience like none other... we have plains, desert, and majestic mountains... sometimes all visable at once... so just about any hiway you choose to drive will take you through spectacular vistas and give you unbelievable opportnities to view wildlife any time of the year...