Officers from South Yorkshire Police are on the island for two weeks to re-appeal to local people and follow up lines of inquiry.
Ben, from Sheffield, was 21 months old when he disappeared on 24 July 1991.
His mother Kerry has always maintained her son, who would now be 26, is alive and was probably abducted.
Ms Needham said she hoped that Greece's statute of limitation laws would mean that people may now come forward in the knowledge they can no longer be prosecuted for withholding information.
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At a press conference, Det Insp John Cousins said: "We specifically want to appeal to anyone who thinks they might be Ben. Did you have blonde hair and blue eyes as a child? Are you unsure of your background?
"I'd like any of those individuals who believe they might be Ben to come forward and speak to us. There will be no pressure on them at all.
"We have Ben's DNA and it will be very easy for us to do a quick test and confirm or otherwise."
Mr Cousins said officers were looking into calls made to the UK's Missing Persons Bureau in 1993 by an anonymous Greek person who had said "Ben was known then as  Andreas and was being cared for by a man called Nikos".
Ben vanished after travelling to the island with his mother and grandparents who were renovating a run-down farmhouse building in the village of Iraklis.
Speaking on the island where her son went missing, Ms Needham said: "The answer does lie on Kos, Ben was abducted from Kos, and someone knows on the island what happened to Ben. I think the appeal this morning by the detectives can bring that person forward.
"Every day is difficult to deal with. It seems, and it's a little bit ironic, but it's nearly 25 years and we're going back to the beginning. I wish this could have been done so many years before but unfortunately it never was allowed.
"I'm just thankful now that the Greek police and the British police are working together. The British government have given funding for this to happen.
"We need to look to the future now, to remain positive and hopefully, we can get some answers.
"There's no word to describe how much this means to me.
"I think we've got the best team of detectives on the case."
Officers travelled to Kos on Sunday, they are carrying out "house-to-house visits" and distributing leaflets and posters during the "next two weeks".
Speaking from the farmhouse where Ben disappeared, Mr Cousins said: "I'm aware there were a number of vehicles and people on this lane on the day that Ben disappeared. I'd also like to appeal to anyone that was here on this day to come forward and speak to us."
Earlier this year, the force secured £450,000 from the Home Office to investigate the disappearance of Ben for another six months.
A DNA test was carried out on a man in Cyprus in an attempt to establish if he could be the toddler, but proved negative in 2013.
In 2012, a police operation focused on a mound of earth and rubble close to where Ben was last seen, but no trace of him was found.
Police have offered a reward of 12,500 euros (£9,800) in the search for missing Ben Needham, who disappeared on the Greek island of Kos in 1991.