Walker died Saturday at age 85.
"We had a great week as a family together, and, in true Olene style, she surprised us and she's gone," said her daughter Nena Slighting. "Bless her heart."
Gov. Gary Herbert ordered U.S. and Utah flags at all state buildings flown at half-staff through Thursday.
"On behalf of all Utahns, we express our gratitude for the sacrifice and leadership of one of Utah's finest public servants," Herbert said in a statement.
Walker, Utah's 15th governor, was the second former chief executive to die this year. Two-term former Gov. Norm Bangerter died in April.
Those two deaths, along with that of former three-term Gov. Cal Rampton in 2007, leave just two former Utah governors alive: Mike Leavitt and Jon Huntsman.
Walker struggled to get the words out in September when asked the question about her legacy, just days before accepting the YWCA of Utah's Lifetime Achievement Award. She was fighting back from a respiratory disease that had put her on oxygen — at the same time pushing through the debilitating effects of two major strokes that occurred in 2014, and which many thought would be the end for her.
But, as she proved so many times in her life, people underestimated her fortitude.
"She is one tough lady," said former House Speaker Nolan Karras, long a political ally and close friend.
That inner toughness in an affable, gentle-mannered woman who liked to greet people with hugs and loved to laugh (often at herself) was part of the persona that prompted some in the rough-and-tumble world of politics to refer to her, dismissively (though never to her face), as "Aunt Bee." The reference to the kind but somewhat befuddled matriarch from Mayberry in the "Andy Griffith Show" also was helped along by the aura of discombobulation that Walker sometimes emitted.
Karras pointed to her habit of losing her purse several times a day or, more than once, opening a desk drawer to discover a check she had forgotten to deposit.
"She kind of gave you this impression of a sweet lady that was sort of disorganized, and then you just watched her clean the clock of people who took her for granted."
Disarming • Former Gov. Leavitt, who recruited her as his running mate in a hard-fought election campaign in 1992, called her manner "disarming." And he said it often proved effective in working through the thorniest skirmishes on Capitol Hill.
"When the legislative process was breaking down on a bill, I often asked Olene to walk upstairs to the Legislature and see what she could do. Inevitably, when she came back, the problem was solved."