Not long ago, the eldest son of President Joko Widodo of Indonesia was running a catering business and a chain of dessert shops. Now he is the symbol of a budding political dynasty and the beneficiary of family maneuvering.

With the help of a high court ruling led by his uncle, the president’s son, Gibran Rakabuming Raka, 36, has emerged as the leading candidate for vice president in next month’s national elections. If his ticket wins, he would become Indonesia’s youngest vice president ever.

The machinations have rattled critics, who warn that Mr. Joko is moving to undermine democratic overhauls that were adopted after decades of dictatorship and that helped Mr. Joko himself win the presidency in 2014.

Three candidates are running to succeed Mr. Joko in Indonesia’s Feb. 14 election, including a former general who is now defense minister, Prabowo Subianto. Mr. Prabowo, who has long been accused of human rights abuses, has lost the last two elections to Mr. Joko.