There are two other Edmondson famiies in the IOM 1901 census, possibly others:

24 Strand St, Ramsey, Maughold RG13/ 5307 f.156
Alfred Edmondson, H, mar, 26, Hairdresser, England
Catherine Edmondson, W, mar, 30, Ramsey
Mary Sayle, servt, s, 15, Housemaid, Ramsey

EDMONDSON, Alfred 4 Oct 1900 R mau 172 CLARKE, Catherine [Brian Lawson's marriage index]

5 Belgravia Ter, Onchan RG13/5302 f.93
Frank Edmondson, H, mar, 25, Electrical Engineer, England
Florence M.A. Edmondson, w, mar, 23, India (British subject)
Connie Kelly, servt, s, 19, general domestic, IOM Rushen

If either of these couples produced an Alice after 1901 would she be too young to be your Gt Grandmother?

Frank Edmondson could possibly be the son of the Oswald and Alice that Brian suggested [and might name a daughter of his own after his mother?]:

(1891) RG12/4686 f.96m p.7 sch.38
Laureston Rd, Onchan parish [follows on after Victoria Rd, Duke's Rd, Prince's Rd, etc]
Oswald R. Edmondson, h, m, 54, Commisioners agent, Manchester
Alice, w, m, 44, Pendleton, Lancs
Lily, d, 19, Lytham, Lancs
Frank, s, 15, Collegian (?), Lytham, Lancs
Helen Townley (?), sister, s, 39, Manchester
Ellen Kinsey, servt, s, 21, Dom. serv., Llandudno

(1881) RG11/3954 f.65 p.39 sch.185
50 Bury New Rd, Salford
Oswald R. Edmundson, h, m, 45, no occupation, Manchester
Alice, w, m, 39, Pendleton
Oswald, s, 15, articled clerk, Lytham
Arthur M, s, 13, scholar, Lytham
Elizabeth, d, 11, scholar, Lytham
Frank, s, 5, scholar, Lytham
Mary Gately, servt, un, 19, dom. servt, Cheshire, Marston

I've got Oswald & family in 1871 and 1861 too if you think this might be your family. Oswald & Alice must have married between the 61 & 71 censuses.

Other details you have that link you to Alice might help identify her for sure.