How would one go about finding information about a parent that left him or her as an infant?

[an unquiet mind]

title pretty much says it all. all i know is the approx year of death and i think the state of residence.

without having to pay, preferably. google hasn't helped much and obituaries aren't that easy to sift through.
i literally have zero information and the death was pre facebook era. the last name is common as $&@#.

is is there any way to find someone based on first name last name job and state of residence?