Day Name Meaning
1) English: from a pet form of David. English: from the Middle English personal name Day(e) or Dey(e), Old English Dæi, apparently from Old English dæg ‘day’, perhaps a short form of Old English personal names such as Dægberht and Dægmund. Reaney, however, points to the Middle English word day(e), dey(e) ‘dairy maid’, ‘(female) servant’ (from Old English d?ge, cognate with Old Norse deigja ‘female servant’, ultimately from a root meaning ‘to knead’, and related to the word for dough), which he says came to be used for a servant of either sex.
2) Irish: Anglicized form of Gaelic Ó Deaghaidh (see O’Dea).
3) Scottish: from an Anglicized form of the Gaelic personal name Daìdh, a colloquial form of David.
4)Welsh: from Dai, a pet form of the personal name Dafydd, Welsh form of David.
Source: Dictionary of American Family Names ©2013, Oxford University Press
You didn't ask for advice, but a much better question would have been "Who were the parents of ____, with your grandfather's and grandmother's name, birth date, birth place, death date and place, and your grandmother's maiden name. A similar question for your maternal grandparents, assuming they too have passed away. You have 8 great grandparents, and only one of them was born with the surname Day.