Sirleaf recently appointed her third son and senior adviser, Robert Sirleaf, as chairman of the state-owned National Oil Co. of Liberia, the BBC said.
He is seeking $8 million in damages from the Independent newspaper over a story about newly discovered oil that appeared with the headline "Sirleaf's oil or Liberian oil?"
The president's son claimed the article was meant to bring public and international ridicule to his office.
The oil company chairman also sued The Analyst newspaper for $2 million and opposition politician Jefferson Kogie for $1 million over an article he wrote for the newspaper.
That article criticized the president's decision to give Robert the oil company job after she had made another son deputy governor of Liberia's central bank.
By publishing the article, Robert Sirleaf said, the newspaper "failed to exercise the degree of responsibility associated with the field of journalism."
The BBC said the president, who took office six years ago, had promised to end "an imperial style of presidency."