Private James Collinson, 17, of Perth, was found dead with gunshot wounds in 2002, with the tragedy the fourth to hit the notorious camp in the space of just seven years.
Human rights lawyers examining files surrounding the shootings say they have already uncovered documents that show evidence was not properly examined and that witnesses lied during the inquests into the deaths of two other soldiers.
Their discovery prompted officials at Liberty to lodge an application for a fresh investigation into Pte Collinson's death.
The soldier's mother, Yvonne Collinson Heath, said yesterday: "This has given me renewed hope; a hope beyond hope that finally this time we'll get to the bottom of what happened at Deepcut.
"With the recent developments in the Hillsborough families' quest for justice it proves that with determination, the truth can, and will, be found out eventually."
Pte Collinson had been on guard duty on the night of March 23, 2002, when he was found with a gunshot wound to the head.
The Army always maintained the four soldiers took their own lives.
Successive governments repeatedly refused demands for a public inquiry, with the official Deepcut Review by leading QC Nicholas Blake also concluding in 2006 that the soldiers most likely killed themselves.
Mrs Heath, 47, of Ellesmere Port, Merseyside, said: "We haven't quite yet abandoned hope of finding out what happened to our loved ones.
"The files - which contain everything from witness statements to the results of forensic tests and crime-scene photographs - have only ever been viewed in part by the coroner and by the Commons Defence Select Committee.
"I don't know what our case notes might hold, but I harbour hope they can raise possible leads to explore."
The Attorney General's office confirmed it had received an application from Liberty.