Your dad is 1/8th Jewish in the same way as I am 1/4 Latvian. It's in the heritage, but not something that is in any way relevant to the way I think, feel, or live my life.
If it wasn't your great-great-grandfather who was Jewish but your great-great-grandmother, and you descended from her in a purely female line, then you would technically be Jewish. Jewish, not 1/8th or 1/16th Jewish.
Orthodox Jews would not consider you, thinker, to be a Jew because your mother was not Jewish.
Reform and Liberal Jews, OTOH, would also not consider you to be a Jew.
But this would be because neither you nor any of your ancestors since that Jewish g-g-gp live as Jews. For Reform and other progressive Jewish groups, faith and practice are almost as important as ancestry.
Judaism is both a religion, a race, a faith, and a way of life. Some people would also say it's a culture, but I'm not sure that Jewish culture - whcih does exist - can stand alone without one of the other aspects. It doesn't have any effect on your citizenship.