Flight MH370, carrying 239 people, went missing on its way from Kuala Lumpur to Beijing, nearly three years ago, on March 8, 2014.
Australia, Malaysia, and China jointly called off a two-year underwater search for the aircraft in January.
Grace Nathan, a Malaysian lawyer whose mother, Anne Daisy, was on the plane, said the families hope to raise $15 million to fund an initial search north of the previous search area.
"We won't start fundraising until we're sure that the governments are not going to resume the search and until the current data has been fully reviewed and analysed," she said at the campaign launch and MH370 memorial event held at a mall in Kuala Lumpur.
The three governments have said they will resume the search if any credible evidence on the whereabouts of the plane emerges.
International experts last year assisted Voice 370, a support group for MH370 next-of-kin, in mounting their own search along the East African coast where debris had been discovered.
"They pinpointed to us accurately where the debris would have made landfall. They've been very helpful both on a personal level and to the investigation," Ms Grace said.
The memorial event, the first held since the search was suspended, featured musical and dance performances, while family members and friends of those aboard made impassioned pleas for the search to continue.
Jiang Hui, whose mother was on the plane, recounted his experience discovering a piece of potential MH370 debris in Madagascar last year.
"I thought it was very miraculous and fortunate when I found the piece of debris that day, but I thought it was useless because this sort of searching activity should have been done by the government," said Ms Jiang, who travelled from China to attend the memorial.
"It should not be us, the family members, who should have been subjected to this pain, to go and face this cruel reality."
Malaysian Transport Minister Liow Tiong Lai, who attended the event, said authorities had analysed 27 pieces of potential MH370 debris along the East African coastline, including two new pieces found in South Africa two weeks ago.
The government has also signed several agreements with countries along the East African coastline to coordinate searches for debris, Mr Liow said.