Darby is a name used for Jeremiah in Ireland.
This is correct and I was about to enter this.
The other point that you may be able to take a
very tentative finding from is that Aussie Darby McAuliffe being the son, reported by Sparrett of Jeremiah and Joanna may be the third son of Jeremiah & Joanna with the two earlier sons named for the father's father and mother's father
Using Irish naming patterns here (from Mrs Google):
"Traditional Irish naming patterns involve naming children after close relatives based on their birth order and gender. For example, the first son would be named after his father's father, while the first daughter would be named after her mother's mother. This practice was prevalent in Ireland, especially from the late 18th century to the middle of the 20th century.
Elaboration:
First Son: Named after the father's father (paternal grandfather).
Second Son: Named after the mother's father (maternal grandfather).
Third Son: Named after the father.
First Daughter: Named after the mother's mother (maternal grandmother).
Second Daughter: Named after the father's mother (paternal grandmother).
Subsequent Children: The pattern continued with the fourth son named after the father's oldest brother and so on, with similar patterns for daughters.
Family History: This pattern can be a valuable tool for family history research, helping to identify relatives and their names.
Modern Usage: While traditional naming patterns are less common today, some families still adhere to them or use variations.
Cultural Significance: The practice reflects the importance of family history and honoring ancestors in Irish culture."
The only warnings I have about using this naming pattern as any more than a rough guide is that the Irish are less likely than the Scots who also have a naming pattern to follow this slavishly.