Cheney’s candidacy was doomed once three-term Sen. Enzi announced his run for re-election. Enzi gets around the state, holding frequent ice cream socials, waiting for everyone else to speak at town hall meetings until he chimes in. He is also a hardworking senator who masters serious public policy issues (note his trenchant criticism of Obamacare before it was passed). Wyoming voters admire him but, more importantly, consider him an old and trusted friend. They were not going to vote him out, even for a challenger who was arguably a bit more conservative, even for a challenger who came from a well-known and still widely admired Wyoming family.
Note that when Dick Cheney was elected to the House of Representatives in 1978, he was not running against a Republican incumbent. It was an open-seat race: Four-term Democratic incumbent Teno Roncalio was not running for reelection. Dick Cheney had been Gerald Ford’s White House chief of staff, but he did not campaign on his laurels. Rather, he returned to the state and campaigned person-to-person, even as he was recovering from one of his early heart attacks.
Had Mike Enzi not run for re-election, Liz Cheney might have won similarly. But she was not going to beat a popular incumbent.