“For the past many months and even years, defendants have dedicated an extraordinary amount of time and energy toward looking for, hacking into, tampering with, manipulating, copying, disseminating and generally obsessing over data that they were given that was taken or stolen from plaintiff’s devices or storage platforms,” including what they claim to have taken from the laptop, the suit says.

Among the laws Mr. Biden says that Mr. Giuliani and Mr. Costello should be held accountable for violating are the Computer Fraud and Abuse Act and California’s Computer Data Access and Fraud Act. The suit did not name a dollar figure that Mr. Biden is seeking, saying only that it was over $75,000.

Ted Goodman, an adviser to Mr. Giuliani, responded in a statement: “Hunter Biden has previously refused to admit ownership of the laptop. I’m not surprised he’s now falsely claiming his laptop hard drive was manipulated by Mayor Giuliani, considering the sordid material and potential evidence of crimes on that thing.”

Mr. Giuliani began trying to promote the contents of the laptop in the weeks before the November 2020 election, hoping to give Mr. Trump an October surprise that could catapult him to victory over Joseph R. Biden Jr. The contents that Mr. Giuliani said came from the laptop included salacious photos of Mr. Biden as well as a trove of emails and text messages describing foreign business contacts and family matters.

As the contents of the laptop became public, Mr. Biden’s aides tried to raise questions about the validity of the material, and dozens of former intelligence officials released a letter that cast information becoming public as having “all the classic earmarks of a Russian information operation.”