Facts and Events
Lookup the ships logs.
Lookup there marriage record to see if any indication is given as to her place of birth or parents.
1860 endownment house record
Also look at the 1860 census record for her husband to see if Mary is also there with him.
Also look at the 1851 uk census for london england for the family.
Mary Smith; Female; Birth: 18 NOV 1823 Newcastle Upon Tyne, Northumberland, England; Death: 04 MAR 1878 , , Utah; Spouse: Joseph Harman; Marriage: 1837 , , England; Film Number: 538240 Page Number: Reference number:
Mary Smith; Female; Birth: 18 NOV 1823 Newcastle Upon Tyne, Northumberland, England; Death: 04 MAR 1878; Spouse: Joseph Harman; Marriage: 1837; Film Number: 538042 Page Number: Reference number:
Mary Smith; Female; Birth: Newcastle, , , England; Death: 04 MAR 1878 <Newcastle, , , England>; Spouse: Joseph Harman; Sealing to Spouse: 01 AUG 1878 Saint George, Washington, Utah; Film Number: 170596 Page Number: Reference number: 3405
MARY ANN SMITH; Female; Birth: 1823; Other: 21 SEP 1844 London, London, England; Batch Number: 6940025 Sheet: 00 Source Call No.: 0087013 Type: Film
Mary Smith; Female; Death: MAR 1878; Film Number: 170542 Page Number: 496-97 Reference number: 9255
Mary Smith; Female; Birth: Newcastle Or,Lynn, , , England; Death: 04 MAR 1878 <Newcastle Or,Lynn, , , England>; Film Number: 170847 Page Number: 274 Reference number: 5318
Mary Smith; Female; Endowment: 14 JUN 1917; Sealing to Spouse: 15 NOV 1917 SLAKE; Joseph Harmon; Other: About 1766 Of, New Marlboro, Berkshire, Massachusetts; Spouse: Joseph Harmon; Sealing to Spouse: 15 NOV 1917 Salt Lake City, Salt Lake, Utah; Film Number: 184603 Page Number: Reference number: 16957
Smith being a common name and Northumberland being a populous place makes this a challenging one to work with. 1780 - 1800 Estimated birth date for Mary parents. Location guess is Northumberland, England. She emigrates from England in 1856(36) arriving in Pennsylvania. I appears they stay in Pennsylvania for awhile. Or at least her daughter stays for awhile. I am not able to find records showing when she comes to Utah. Her daughter immigrates to Utah in 1862. In her daughters history it mentions the following.
The family began preparing for the trip to Utah soon after Baptism. Mother (Mary) and daughters, Mary Sarah and Elizabeth (of 49 Great Cambridge St. Hackney Road, London, England sailed from Liverpool, (notification date 15th November 1855) on the ship Emerald Isle. They arrived in Philadelphia, Pennsylvania in 1856 where the family lived for a number of years. Father (Joseph) and son George were not listed among the passengers but they most likely were working their way over for their passage on the ship.
Things to check.
Check the 1860 Pennsylvania census records.