Latham’s birth is well before the start of statutory records (1864) and so you will need to rely on church records to trace him. You haven’t said what denomination he was. Do you know?
I looked at the 1831 census for Templemore, and there were 4 households headed by a John Brown. 1 in Edenballymore was Presbyterian, 2 in Killea & Priest Lane that were RC and 1 in Fountain St that was Church of Ireland. Only the heads of household were named in the 1831 census so I can’t tell whether any had a son named Latham. However Latham is a pretty unusual name. I did search the census and found just one Latham in the county. He was Latham Orr and also lived in Edenballymore. So I wonder if there is any family connection?
The Church of Ireland records for the parish start in 1642; there are several Presbyterian churches but the only one with records for 1823 is 1st Derry (they start in 1815) and the RC records start in 1823 so you might just scrape in, if that’s the denomination. I searched the records on rootsireland but did not find any record of Latham’s baptism.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society are running an Ulster DNA project and can offer FTDNA testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).