I cannot confirm that this William is son of Melvin and Eve White except by excluding all the other William White's in 1850 and 1860 census. These coincidences make the connection likely though:
1) born 1831 NY
2) 1856 married in Erie Co, PA which is next to Chautauqua Co, NY
3) named a daughter Eva.
4) millitary service & died in Civil War. Claims he was "William H White" here volunteer placed in "3rd Regiment, Pennsylvania Heavy Artillery"
http://www.researchonline.net/pacw/index/index933.htm
and here - http://www.nps.gov/civilwar/search-soldiers-detail.htm?soldierId=7399ACDE-DC7A-DF11-BF36-B8AC6F5D926A
M554 ROLL 130 alternately lists "William W White" which is how the pension application was also labeled.