* Davis trailing Republican rival as state party meets
* "Battleground Texas" group aims to mobilize non-voters
By Jon Herskovitz
DALLAS, June 29 (Reuters) - For one day a year ago, Wendy Davis became the brightest star in the U.S. political universe when she donned pink tennis shoes and launched a one-woman, 10-hour filibuster against abortion restrictions that brought her international attention.
Now she is battling to revive a seemingly stalled campaign to become the first Democratic Texas governor in more than 20 years by winning over frustrated Republicans and motivating enough voters who would otherwise spend election day at home to find a few minutes to vote.
State Senator Davis, 51, came into the Texas Democratic convention in Dallas over the weekend with surveys showing her 10-13 percent points behind the Republican nominee, Attorney General Greg Abbott, 56, and failing to close ground.
Davis, with an inspiring life story going from a single mother in a trailer park to a Harvard Law School graduate, has portrayed Abbott as part of a 'good old boys' network more interested in enriching each other than helping voters.
"I'm running because there's a moderate majority that's being ignored - commonsense, practical, hardworking Texans whose voices are being drowned out by insiders in Greg Abbott's party, and it needs to stop," she told the convention on Friday.
But as Mark Jones, a political science professor at Rice University in Houston, said: "Texas is difficult terrain for any Democrat, let alone a Democrat who rose to prominence on an issue such as abortion that is associated with the liberal wing of the Democratic Party."
Davis, who this month reshuffled campaign brass, has also taken hits when it was found she embellished parts of her biography.
Despite this, she remains a prominent candidate who can raise funds among major donors in Hollywood and Washington, D.C.
For her staunchest supporters, the key to victory rests with Texans being targeted in one of the largest, state-wide grassroots campaigns in U.S. political history.
Alumni from President Barack Obama's 2012 campaign quietly have built a Democratic political army in Texas, where gun-rights advocates brandish semi-automatic rifles on city streets and pickup trucks bear "SECEDE" bumper stickers.
The group called Battleground Texas, started about a year and a half ago, has enlisted about 20,000 volunteers who have made 2 million phone calls and house visits among voters.
"People are hungry. They see the opportunity and they want to take it," said Jenn Brown, the group's executive director.
In a state as large as Texas, adding a few percentage points with groups that typically favor Democrats but have not turned out in high numbers could mean all the difference.
According to a poll by survey group Latino Decisions, 39 percent of eligible Hispanics cast ballots in the November 2012 election in Texas, while 61 percent stayed home. The numbers were almost reversed for non-Hispanic whites.
By 2030 Latinos, who typically support Democrats, will be the majority in Texas and could turn the state blue. Texas could then join populous California and New York among Democratic strongholds, with the three states securing the party nearly half the electoral votes needed to win the White House.
Democrats have also become emboldened by the strength of U.S. Senator Ted Cruz of Texas and his no-compromise tea party allies, seeing them as driving moderate Republicans into their tent and antagonizing Hispanic voters with their hardline stance on immigration.
"We have seen Republicans consistently move further to the right and as they are excluding people from the promise of Texas, our job is to speed up those changes," said Will Hailer, executive director of the Texas Democratic Party. (Additional reporting by Marice Richter; Editing by Tom Heneghan)