Genealogy /
THE MEANING AND RESEARCH OF MY POLISH LAST NAME, SURNAME? [4500]
It is a Lancashire name as this map shows.
surname.sofeminine.co.uk/w/surnames/surname-baines.html
This interesting name has three possible origins. The first from the Gaelic "ban" meaning "fari" or "white" as in Ewin Bayne alias Quhyte recorded in Perth, Scotland in 1623. Secondly from the Olde English "ban" meaning "bone", which later became "bon" and survives as a nickname "Bones" in the Midlands and Southern England and thirdly from the Olde French "bain", bath, which possibly denotes occupation as an attendant at the public baths. It is also possibly a locational name from France a village near Bayeux in Normandy.Read more
or
Anglicized version of Welsh "ab Einws" (son of the anvil) ; or English name associated with Bainton in Yorkshire ; or Scottish variant of Bain (white , fair) ; or from Scottish "Beathan" (life)
Unless of course it was changed from something else, like Banaszek or BanaĆ, in order to anglicise it.