Barton, the township, overlapped with most of the old (pre-amalgamation) city of Hamilton and was fully annexed by 1960. Barton, the street, was probably named for it.
From the Hamilton Public Library's "Local History & Archives" --
https://lha.hpl.ca/articles/chronolo...ships-hamilton
1816
On March 22, 1816, a new district was created out of the old divisions of the Home and Niagara Districts. The new Gore District was named after Sir Francis Gore (1769-1852), Lieutenant-Governor of the colony. This district consisted of the two counties of Wentworth and Halton. At this time the boundaries of Wentworth County were not yet permanently fixed so it was determined that Wentworth would encompass the Townships of Saltfleet, Barton, Binbrook, Glanford, Ancaster and “so much of the County of Haldimand as lies between Dundas Street and the Village of Onondaga”.
1960
The last of Barton Township was annexed by the City of Hamilton and the township ceased to exist.
Excerpts from the same site, next page --
https://lha.hpl.ca/articles/chronolo...ton?page=0%2C1
Township Histories
Barton
The township of Barton was named after Barton upon Humber in Lincolnshire, England, opposite the city of Kingston upon Hull. This township was established originally in Lincoln Township in 1791 and became part of Wentworth County in 1816. As the city of Hamilton began expanding east and south it began annexing portions of Barton Township and completed the annexation of the township in 1960 at which time the name disappeared from the map.
Hamilton
The city of Hamilton was named after George Hamilton (1788-1836) who set out the first plan of the town about 1820. The area was first known as The Head-of-the-Lake for its location at the western end of Lake Ontario. Hamilton was incorporated as a town in 1833. Hamilton was incorporated as a city in 1846. As Hamilton expanded it annexed portions of Ancaster Township to the west, portions of Saltfleet Township to the east and portions of Barton Township to the east and south, eventually annexing all of Barton Township by 1960. When Regional Government was implemented in 1974 the city of Hamilton became part of the Regional Municipality of Hamilton-Wentworth.