WASHINGTON — For Lt. Gen. Michael T. Flynn, who is President-elect Donald J. Trump’s choice for national security adviser, pushing conspiracy theories about Hillary Clinton is a family affair: Both he and his son, Michael G. Flynn, have used social media to spread fake news stories linking Mrs. Clinton to underage sex rings and other serious crimes, backed by no evidence.

The Twitter habits of both men are attracting renewed attention after a man fired a rifle on Sunday inside Comet Ping Pong, a Washington pizza restaurant that was the subject of false stories during the campaign tying it and the Clinton campaign to a child sex trafficking ring.

Well before he joined the Trump campaign, the elder Mr. Flynn, 57, a former director of the Defense Intelligence Agency, pushed unsubstantiated claims about Islamic law’s spreading in the United States and about the attack on the American diplomatic compound in Benghazi, Libya. But in his emergence this year as the angry former general out to help Mr. Trump clean up Washington, Mr. Flynn added wild stories about Hillary Clinton to his stock of unproven tales.

Six days before the election, for instance, Mr. Flynn posted on Twitter a fake news story that claimed the police and prosecutors in New York had found evidence linking Mrs. Clinton and much of her senior campaign staff to pedophilia, money laundering, perjury and other felonies.