TAMPA, Fla.—Hillary Clinton named Virginia Sen. Tim Kaine to be her running mate Friday night, turning to a seasoned politician from a battleground state and a figure long seen as the favorite to complete the Democratic presidential ticket.

“Tim is a lifelong fighter for progressive causes and one of the most qualified vice presidential candidates in our nation’s history,” Mrs. Clinton said in an email to supporters a day after GOP nominee Donald Trump scorched her in a speech to the Republican National Convention. On Twitter,...