She served in Grantham from early 1915, in the Women's Police Service until 1918, when she left due to ill health.
There is a commemorative blue plaque on Edith Smith Way, off St Peter's Hill, to mark her achievement and place in history as the first WPC to have the power of arrest.
South Kesteven District Council Cabinet Member for Communities, Health, Wellbeing and Skills, Councillor Jacky Smith, and fellow District Councillor for Grantham, Ray Wootten, are each donating £100 to the appeal.
Cllr Smith said: "Edith Smith served Grantham and its people before moving on to nursing in Cheshire and it is only right that her pioneering work is recognised at her final resting place."
Cllr Wootten, a retired police officer who is vice chairman of the Lincolnshire police and Crime Panel, said: "Her dedication, energy and courage made great strides for women at a time when they weren't even paid members of the force."
On leaving Grantham, she made her home in Halton Almshouses, in Runcorn, Cheshire, where she eventually took her own life, resulting in her burial with no headstone in a grave shared with her infant grandchild.
Chairman of Grantham Civic Society, Courtney Finn, said: "The family felt great shame and so no gravestone was ever put at her grave and it has remained unmarked ever since.
"The Birkenhead suburb of Oxton became aware of their famous resident as a result of our Blue Plaque in Edith Smith Way, and Merseyside police wish to further honour her memory and role as the first warranted female police officer by marking her resting place with a gravestone."
Merseyside Police have created a Just Giving appeal online with a donations target of £2,000 at justgiving.com/campaign/rememberwpcedith
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