Use the undocumented Extrusion option:
ContourPlot3D[x*z - y^2 == 0, {x, -1, 1}, {y, -1, 1}, {z, -1, 1},
Mesh -> None, Extrusion -> 0.1]
Another option:
ParametricPlot3D[{x, y, y^2/x}, {x, -1, 1}, {y, -1, 1},
PlotRange -> {-1, 1}, PlotStyle -> Thickness[0.1]]
