The person's appearance is going to be more and more variable for each generation of "dilution," since we're talking mostly white with a little black.
Of course, someone with one black parent will have half black, half white genes. They'll probably look like you think they'd look.
But that half-black person's gametes don't "know" to divide their heritage up 50/50. A half-black person's child is a quarter black if you're looking at a diagram, but that half-black person's sperm or egg that made half of the quadroon child could have been 99% Grandma DNA, 1% Grandpa DNA. Or vice versa, or 50-50.
e.g.. My mother is by definition 50% Grandpa and 50% Grandma. But there's no way to know without extensive testing if I am 25% (maternal) Grandpa and 25% Grandma, or if I got nearly all of Grandma's genes via my mother and almost no Grandpa. Without DNA testing, that's just aunts saying "aww she has Dad's exact frown."
If you have anyone's family photo albums (and a key to who's who), you can see this play out on the family resemblance level.