She married Geo. Cowden on 9 Feb. 1849 in St. George's, Paisley, Renfrew, and her name is given there as Hamilton (see attached source document). However, her death certificate (attached source document) gives her father's surname as Patterson, and she apparently also signed the name Patterson on documents (I have an image of that, but don't know the source).
Was she actually the blood daughter of John Patterson and Agness Caulderwood Patterson, who married 10/2/1815 Neilston, Renfrewshire? It seems probable that Agnes Caulderwood was her blood mother, since she herself gave her own daughter Agnes the second name "Caulderwood" [@Agnes Caulderwood Cowden].
Regarding Agness Caulderwood Patterson, the 1841 Census of Seedhill, Paisley Low, Renfrewshire records her living with three Patterson children, as well as three much younger children with different surnames.
Agness Calderwood 46 Not born in county
William Paterson 25 Weaver ditto
John Paterson 23 Blacksmith ditto
Isabel Paterson 21 Cotton Throwster Born in County
Agness Hamilton 14 ditto
Jean Leitch 7 ditto
Margrat Craig 4 ditto
Two of those minor children named above, Agnes Hamilton and Jean Leitch, might have been related, because there is a record of an Agnes Hamilton born to James Hamilton and Mary Leitch in 1827. The question remains, who exactly are the parents of these last three children?
For a discussion on her name/paternity, see: https://www.rootschat.com/forum/index.php?topic=770328.0.