The Boeing 777 jet with 239 people on board vanished on March 8, 2014 during a routine flight from Kuala Lumpur in the world's greatest aviation mystery.
An official report released in July following a lengthy investigation and years of fruitless searching gave no new clues about why the plane disappeared, sparking anger among relatives.
V.R. Nathan, whose wife Anne Daisy was on the ill-fated jet, told AFP the debris consisted of five small plane parts found off Madagascar.
They were turned over to Malaysian Transport Minister Anthony Loke on Friday at his office in the administrative capital Putrajaya outside Kuala Lumpur.
"Five new pieces of debris have been recovered off the coast of Madagascar, including one piece that has part of a label still readable," Nathan told AFP, adding the items were discovered by fishermen.
"We want the government to continue searching for these debris and piece them together like a jigsaw puzzle so that we can get some clue as to what happened to the plane."
His daughter Grace told reporters the items were found between December 2016 and August 2018, adding that "this... offers a fresh ray of hope to all the relatives."
Only three confirmed fragments of MH370 have been found so far, all of them on western Indian Ocean shores, including a two-metre (six-foot) wing part known as a flaperon.
Malaysia's new government, which took power in May, has said the search could be resumed but only if new and compelling evidence comes to light.
Loke, the transport minister, said the government will "immediately verify the items" ranging from almost 60 centimetres (two feet) in length to palm-sized.
"We will need some credible leads before we reopen the search," he told reporters.
T. Mohan, one of the experts who took part in the investigation, told AFP that one of the items "is a floor panel of a Boeing aircraft."
The disappearance of MH370 triggered the largest hunt in aviation history.
However searchers scouring a 120,000-square kilometre (46,000-square mile) area of Indian Ocean have failed to locate the missing plane.
An Australian-led hunt was suspended in January last year.
US exploration firm Ocean Infinity resumed the search in a different location at the start of this year on a "no find, no fee" basis, using high-tech drones to scour the seabed.
But that search was also called off within months after failing to find anything.