Hi,
Thank you for your message.
The fact that you know she came from Tartaraghan is a good starting point! Most Irish record sources-church records, land records, census records- are based on specific locations, usually a parish.
Do you know her religious denomination? You could try searching for her birth/baptismal records in the hope that her parent's names are listed.
Most Catholic records are held locally. One website which may prove useful is the irishtimes. Here, they give an overview of what records are available in specific parishes. For Tartaraghan/Loughgall follow this link:
http://www.irishtimes.com/ancestor/fuses/rcparishmaps/index.cfm?fuseaction=showidrecords&CityCounty=Armagh&parish=Loughgall&churchid=227
For Church of Ireland records check: http://ireland.anglican.org/about/42 and http://www.nationalarchives.ie/.
You can also do a search of Griffith's Valuation. This was a property valuation survey carried out in Ireland between 1848-1864. I did a search for Burgess in Co. Armagh. 9 results came back two of which were from the parish of Tartaraghan. It is possible that these may be connected to Patience. Although, the survey doesn't offer any genealogical information you can perhaps use these two names as a starting point. You can see the results here:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNameSearch&Submit.x=62&Submit.y=20&Submit=Submit&familyname=burgess&firstname=First+Name&baronyname=&countyname=ARMAGH&unionname=&parishname=
You could aslo search the 1901/1911 census records to see if you can find any family members that may have remained in Ireland:
http://www.census.nationalarchives.ie/search/
Some other websitse which may help you are:
The National Archives of Ireland http://www.nationalarchives.ie/genealogy1/genealogy-records/introduction/
The National Library of Ireland http://www.nli.ie/en/family-history-introduction.aspx
The National Archives UK ? genealogy search: http://www.nationalarchives.gov.uk/records/looking-for-person/
The Public Records Office of Northern Ireland http://www.proni.gov.uk/index/family_history.htm
Family Search: https://familysearch.org/
US National Archives/Immigration info: http://www.archives.gov/research/immigration/
Genealogy Links: http://www.genealogylinks.net/uk/ireland/armagh/index.html
I hope that this information helps. Remember to post any new information you find. The more information you post, the more likely it is that one of our volunteers will be able to advise or assist you. Also include information concerning which sources you may have already used so others may further your search.
Kind regards,
Genealogy Support