Veronica: Here is the 1880 census for Ashlie; they used this name in the family (Ashlie, Ashley) for males. So maybe the "cousin" used ASH for a masculine sounding name?
Henry WILKINS Male age 49 NY Farmer NY NY
Eunice (Winch) WILKINS Wife M Female W 50 NY Keeping House NY NY
Linnie WILKINS Dau S Female W 25 NY At Home NY NY
Pheba (Phoebe) WILKINS Dau S Female W 23 NY NY NY
Ashlie WILKINS Other S Male W 4 NY NY NY