Charity worker Sarah McGuire, 38, who was only 10 years old when Ms Doyle died, told of the discovery a week after the victim's naked body had been found near her Greenock home in June 1986.
She recalled the incident at the High Court in Edinburgh yesterday where John Docherty, 49, is on trial for the 27-year-old murder.
Ms McGuire told how she and a friend regularly visited the James Watt library in Greenock's Union Street. They cycled there on June 9 1986, took their bikes up the steps to the front door and dumped them, leaving the library again 20 or 30 minutes later.
"I noticed a handbag," said Ms McGuire. "I remember it being navy blue. It was what you would refer to as a clutch bag.
"It didn't have a strap. It wasn't very big and it was burned, partially.
"There was a substantial part of the blue bag left but a section of it had been burned."
She said they told her friend's mother what they had seen and police took away the handbag.
Ms McGuire told the trial she did not remember seeing the bag on her way in to the library. Mr Docherty, of Hunters' Quay Holiday Village, Dunoon, denies murder. He claims that, when he is alleged to have stripped and strangled Ms Doyle, 16, he was at home with his parents, who are no longer alive.
Mr Docherty has also lodged a so-called special defence of incrimination claiming the culprit might be one of 41 people named in police files on their investigation into the murder.
The charge alleges that, on June 2 1986 in a lane near Elaine's home in Ardgowan Street, Greenock, he struck her on the head and either removed or compelled her to remove her clothing.
The charge goes on to allege that Mr Docherty forced Ms Doyle to the ground, pushed her face into the ground, sat or knelt on the teenager then placed a ligature round her neck and strangled her. Mr Docherty also denies stealing a handbag from Ardgowan Street on the same date.
He further denies a charge of attacking another woman, Linda Hargie, on various occasions between 1990 and 1995 at an address in Anne Street, Greenock by seizing her and pushing her and punching her on the head.
The trial continues on Tuesday as Monday is a public holiday in Edinburgh.