I think that the Fanny in the 1901 census died on 3.11.1905. Her husband was named John. There is only 1 Hazelton household in Lisgallon in 1901 & 1911 and Fanny died there. Informant was her daughter Jane Dickson.
Some probate abstracts for the family taken from the PRONI wills site:
Probate of the Will of John Hazelton late of Lisgallon County Tyrone Farmer who died 18 October 1899 granted at Armagh to Thomas Bloomer and George Little Farmers.
The above will names John’s wife as Fanny. It mentions children Abraham, Jane Dickson (likely to be her married name), Robert, Edward, Thomas, Ellen, Margaret Marianne & Simon. The will is on-line on the PRONI wills site.
Hazelton Robert of Lisgallon county Tyrone farmer died 10 August 1937 Probate Londonderry 5 July to Hugh John Bennett rate collector and John Taylor retired farmer. Effects £108 7s. 3d.
Hazelton Minnie of Lisgallon Greystone Dungannon county Tyrone widow died 28 January 1943 Administration Londonderry 19 June to Sydney Hazelton surfaceman. Effects £215 9s. 2d.
The above 2 wills are not on-line but should be in PRONI in paper format.
This is Jane in the 1901 & 1911 censuses:
Tradition was to marry in the bride’s church, so if you are looking for Fanny’s baptism try Drumcree Church of Ireland as that’s where she married in 1850.
Jane Eliza Hazelton was 28 when she married on 18.11.1892 at Derrygortreavy Church of Ireland. She was living in Cullenrammer and was a grocer. She was a spinster.