DHAHRAN, Saudi Arabia—Saudi Arabia’s state oil company is working to turn itself into an innovation powerhouse while its much-delayed plans to go public remain on hold.

The U.S. Patent and Trademark Office granted Saudi Arabian Oil Co., better known as Aramco, 230 patents last year, four times as many as in 2013 when it racked up just 57. That placed Aramco third among oil-and-gas exploration and production companies that year, far behind Exxon Mobil Corp. but close to Chevron Corp.

Aramco’s...