Some, however, are not excited about her candidacy.
Utah Education Association President Sharon Gallagher-Fishbaugh said she'd rather see someone with experience as a public school educator take the lead. Lockhart lists her profession as homemaker/registered nurse.
It's critical for a superintendent to have experience with school operations, Gallagher-Fishbaugh said, as well as "experience in a classroom as an educator, as a principal and administrator, as someone who understands curriculum, someone who understands what teaching is like."
She said Utah has seen "example after example of people who are not in education micromanaging and imposing policy on the education system that has failed."
Gallagher-Fishbaugh cited school grading as an example. Utah schools now receive grades of A-F based on test scores and other measures under legislation passed in 2011.
The legislative relationship • Lockhart was among the many lawmakers who supported that initiative, which proponents say provides transparency and accountability.
This past legislative session, Lockhart also pushed to put a digital device in the hands of every Utah student. But that measure, priced at hundreds of millions of dollars, failed, with many lawmakers saying it was simply too expensive.
Lockhart worked on that bill with Rep. Francis Gibson, R-Mapleton, who said Friday he thinks she would do a good job if chosen as state superintendent. Her understanding of the inner workings of the Legislature is an important quality, he said, given how closely the state superintendent must work with lawmakers, who set the state's education budget. Gibson also leads the House Education Committee.
"Do I think she'll have a lot to learn? Yes, but the one thing I've noted working with Becky is she doesn't pretend to know everything, that she'll surround herself with people with good content knowledge," Gibson said. "She may not be an educator, she doesn't have a teaching degree, but she's a leader."
Lockhart did not respond Friday afternoon to additional questions about her possible goals as superintendent.
Changing the governor's race • Losing one of his potential top rivals for the governor's office doesn't faze Gov. Gary Herbert, said his spokesman, Marty Carpenter.
Herbert will stay out of the selection of the new superintendent and won't support any of the candidates, Carpenter said.
"The field [for the governor's race] as it may shape up or may not shape up for 2016 is far less a consideration for the governor at this point than [for] a lot of people who follow politics closely," Carpenter said. "If everyone who had been rumored to run in the governor's race decides to jump in, it wouldn't change his position on this. … It's not his decision to make, so he's just staying neutral."
Utah State University political science professor Damon Cann said Lockhart may be positioning herself to be a leader on an issue where the governor has been criticized by conservatives the Common Core education standards and still could run against Herbert, although it is unlikely she would do so.
"Becky Lockhart was rattling cages and making some noise and all that, and she seemed to really be the one who had the most credibility to go after him," Cann said.
"At this point, it's hard to imagine another challenger coming out of the woodwork who could be as strong as Becky Lockhart," he said. "… There would need to be a major misstep before Herbert would be vulnerable."
The needed skills • Dawn Davies, Utah PTA president-elect, said Friday the PTA does not take positions on candidates, but it has sent the state board a list of qualifications it would like to see in the next state superintendent.
They include: knowledge of public education, rules, laws, regulations and processes; knowledge and ability to testify and communicate clearly at the Legislature; and the ability to develop relationships with employees and the community, among other things.
Judi Clark, executive director of Parents for Choice in Education, also didn't comment specifically on Lockhart's application. But she said she's encouraged by the state school board's efforts so far pointing to the board's decision this week to appoint Joel Coleman as interim superintendent.
That shows "they're interested in finding the right person, not the person who's available right now," Clark said.
Coleman is a former state school board member and superintendent of the Utah Schools for the Deaf and the Blind. His appointment drew some controversy, with the board voting 9-6. Some members were concerned about a pending lawsuit his wife filed against education officials and the state charter school board. Others decried Coleman's lack of experience as a public school teacher.
Coleman has spent his career as a seminary teacher and administrator for The Church of Jesus Christ of Latter-day Saints.
What's next? • Board vice chairman Dave Thomas said the board may vote on a new superintendent at its Oct. 3 meeting.
Before then, he said, the board will select semifinalists, who will be vetted by board members and representatives of various groups, including district superintendents, lawmakers, the PTA and school boards. The board's search committee will then narrow the list to finalists, who will be interviewed by the full state board.
That final vote must be public.
But Thomas said the board hasn't yet decided how much of the rest of the process will be public. During the last superintendent search, finalists' names were announced and their interviews were public.
The board will conduct a nationwide search with the help of a Salt Lake City-based firm. Thomas said he believes it's important to choose the best person for the job.
"That doesn't mean we're looking for any specific individual from outside education nor does it mean we're looking for anyone specifically within public education," Thomas said Friday. "We're looking for well-rounded individuals who can best move forward the priorities of the state board of education."
Matt Lyon, executive director of the Utah Democratic Party, said the school board should not be swayed by Lockhart's connections.
"Lockhart's qualifications, her job history, should be the consideration to look at and not her status in Utah politics," he said.
The opening comes amid allegations of dysfunction among board members and the departures of both Superintendent Menlove and deputy superintendent Brenda Hales. Earlier this month, Hales stopped working, saying she would use vacation and other leave until her retirement becomes official at the end of the year.
Shortly after that, Menlove, who had previously announced his retirement, decided he would leave in coming weeks rather than wait for a permanent replacement as he once planned.