It might be worth following up on possible siblings, e.g. Catherine.

In 1865, a Catherine Henney married Andrew Healey at Dalry. If you look at this marriage record, you will get the names of Catherine's parents.

Andrew Healey was considerably older than Catherine and had a previous wife (Margaret) and children. The Healey family are in all subsequent Census records.

In 1861, Catherine Hainey, age 18, born Kilbirnie, was an Out Door Worker, boarding with the Higgins family at No. 49 Swinlees Row, Dalry.
In the 1861 Census, Andrew Healey (Healy, Haley), at age 38, was enunmerated at No. 53 Swinlees Row. His occupation: Drainer. His wife then was named Margaret, and there were 5 children, the 2 youngest named Margaret and Jane.
In the 1871 Census, Andrew Healey is now shown as age 40, and his wife is now Catherine Haley, shown age 32, born Dalry.
Children with them are:
Margaret Haley, 15
Jane Haley, 12
Elizabeth, 5
Andrew, 3
Mary, 1