Weeks, who was based in the West Australian city of Perth, was one of 239 people on board the Boeing 777 Kuala Lumpur-Beijing flight when it disappeared on March 8, 2014.
Next-of-kin had started to file lawsuits over the jet's disappearance as a two-year deadline approached last week, with some hopeful the court scrutiny could shed light on what happened to the ill-fated flight.
Under international agreements, families have up to two years to sue over air accidents.
They were seeking compensation for "personal injury, loss and damage" suffered as a result of "sudden shock" and "mental harm" after the disappearance, the Times said.
Also Read
"The cause of the plaintiff's pain, injury, loss and damage was negligence of the defendant," the newspaper cited the writs as saying.
A fifth writ was filed on behalf of Weeks' two children, the report added, without disclosing the amount of damages sought.
Malaysia Airlines also faces legal action elsewhere. In Beijing, relatives of a dozen Chinese passengers filed suits against the airline, Boeing, Rolls Royce and others last week.
The latest legal claim in Australia came as a piece of debris found by a South African holidaymaker in Mozambique in December was to be sent to Australia for analysis, South African officials said Friday.
Other pieces of debris found earlier this month have yet to be confirmed as from the missing jet.
No crash site has been located. So far, only a wing fragment that washed ashore on the Indian Ocean island of Reunion last July has been confirmed as being from the plane.