Daveey - I've no information, only a suggestions, which are probably well worn ground, but in case:
Local newspapers for the Fleetwood area - were there any details at the time of the gift/launch/arrival of "Edith" that might help give information on where she was built?
Similarly you could try papers covering New Brighton for reports on build/repair of liefboats in general or of "Edith". Some libraries have online access to the Gale newspaper collection - mine doesn't but of your area doesn't some other RootsChatter may be able to do a search. Otherwise your local history library may be able to help with old newspapers.
Finally the RNLI Heritage Trust may be able to help with sources or suggestions or have some relevant information
http://www.rnli.org.uk/who_we_are/the_heritage_trustBest wishes for the anniversary!