Japan’s governing party will convene on Saturday for a critical election as it looks to revive its sagging fortunes, a vote that could pave the way for the country’s first female prime minister or perhaps its youngest leader in 140 years.

The Liberal Democratic Party is set to choose a successor to Shigeru Ishiba, Japan’s prime minister, who announced last month his intent to resign after less than a year in office. The party, which has ruled Japan for all but five of the last 70 years, performed badly in July elections, leaving it in the unusual position of being a minority in both houses of the Diet, Japan’s Parliament.

Five candidates are vying for the L.D.P.’s top post, and the winner is expected to be named prime minister in mid-October during an extraordinary session of Parliament.

Japan’s next leader will face a series of challenges, including a strained relationship with the United States; rising food prices and stagnant wages; a backlash against immigration; and fresh questions about the L.D.P.’s image and influence amid scandals and the rise of right-wing politicians.

“The L.D.P. needs to rejuvenate its image,” said Shigenobu Tamura, a political analyst in Tokyo and a former L.D.P. official. “Society wants a change now.”