The 62-year-old's son, Steve, said his father is an experienced runner.
Search teams, including police, fire crews and tourists, are following up a possible sighting, he added.
Mr Lawton has not been seen since the latter stages of the competition.
His wife Lynda is in Greece helping with the search effort.
The British embassy in Athens has also been informed and is assisting.
Steve Lawton said: "(We are following) a small lead from a lady who thought she saw him at a particular area.
"Now, obviously he was still running at that point but it moves the search slightly forward.
"Every single pair of eyes or feet really helps. It's all a case of looking at every eventuality."
He said the terrain was "incredibly mountainous" with deep gorges and sheer drops.
"My dad knew what he was doing," he added.
"He was used to fell running and that terrain and when I'm going through the route, by and large, unless you are very disorientated you could find ways of following that route.
"There are lots of places where you could slip and sadly some those areas potentially could mean a very, very significant fall.
"There are areas where you could fall and get totally immersed in foliage and bracken and trees."
Keeping the momentum going for the search now his father has not been seen for five days was the most important thing, he added.
"Whatever the statistics are, I think to us, are irrelevant because it's just about if there is hope then it is about continuing the search."