Dr Jim Swire, speaking at a screening of a new documentary on the original investigation, said they were tired of waiting on the Scottish Government.
It will now be up to the Scottish Criminal Cases Review Commission, which previously identified six grounds for appeal, to decide whether it is in the public interest to refer the case back to the Court of Appeal.
The Scotsman revealed last year that British families were considering making this move.
They had delayed to see if the Scottish Government would launch an inquiry, or if the family of the late Abdelbaset Ali Mohmed al Megrahi – the only person convicted of the bombing – would trigger an appeal.
However, Dr Swire said more than 25 years after Pan Am 103 exploded over Scotland, killing 270 people, the families had “decided enough is enough”.
Speaking at a screening of al-Jazeera’s Lockerbie: What Really Happened? in the Scottish Parliament yesterday, Dr Swire said: “The content of today’s production seems to match, very closely, the information we have as relatives. But there’s limits to what we can do.
“The Scottish Government has repeatedly been asked to launch an inquiry into this and we know they have the power to do so. Some British relatives have decided enough is enough and we will be applying within weeks for a further appeal.
“That should raise this to another level.”
Dr Swire’s 23-year-old daughter, Flora, was killed in the Lockerbie tragedy.
The documentary once again pointed the finger at Iran, saying it funded the attack, which was carried out by the Popular Front for the Liberation of Palestine General Command (PFLP-GC)).
It says it has seen US intelligence cables, which read: “The execution of the operation was contracted to Ahmad Jabril, Popular Front for the Liberation of Palestine General Command (PFLP-GC) leader. Money was given to Jabril up-front in Damascus for initial expense. The mission was to blow up a Pam Am flight.”
The documentary heard from CIA agent Robert Baer, who said an “executive decision” was taken, emanating from the White House, to focus on Libya.
It also heard from Jessica De Grazia, author of the Bird Report, commissioned by Megrahi, and former senior Iranian intelligence official, Abolghasem Mesbahi. He claimed the Lockerbie bombing was retaliation for the shooting down of Iran Air Flight 655 by the USS Vincennes in July 1988, which killed 290 people.
Mr Mesbahi said: “Iran decided to retaliate as soon as possible. The decision was made by the whole system in Iran and confirmed by Ayatollah Khomeini. The target of the Iranian decision-makers was to copy exactly what’s happened to the Iranian Airbus.”
Mr Baer said: “The Iranians went to the PFLP-GC, the break-off faction and said, alright you guys know how to bring down planes, bring down five, five civilian airliners.”
The Crown Office, which is still treating Lockerbie as a live investigation and has signed information-sharing agreements with Libyan prosecutors, dismissed the documentary as old news.
“There is nothing new in these claims and some of the accounts of the evidence reported are inaccurate,” a spokesman said.
“The only appropriate forum for the determination of guilt or innocence is the criminal court, and Mr Megrahi was convicted unanimously by three senior judges.
“His conviction was upheld unanimously by five judges, in an Appeal Court presided over by the Lord Justice General, Scotland’s most senior judge.
“As the investigation remains live, it would not be appropriate to offer further comment.”