I'd like to simply add a nice thick point in $(0,0,0)$ to Plot3D[(y^2/x), {x, -3, 3}, {y, -3, 3}].

The solutions I came across to add anything to Plot3D, mostly involved MeshFunctions. However I don't see how I could manipulate the mesh to give the desired result.

Any help would be appreciated.

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Use Show and add Graphics3D[{PointSize@.5, Point[{0,0,0}]}]. And welcome to Mathematica.SE :) –  Kuba Jan 30 '14 at 21:04
closely related: add point to existing graphic –  Kuba Jan 30 '14 at 21:11

Show[
Plot3D[(y^2/x), {x, -3, 3}, {y, -3, 3},
PlotStyle -> Opacity[0.3]],
Graphics3D[{Red, PointSize[0.1], Point[{0, 0, 0}]}]]

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This may be an overkill but it is useful (no joke!) cause when you export points and lines they tend to look odd the thicker they are. You can fake a red point by re-difining your ColorFunction like so:

With[{colf = Function[{x, y, z},
If[Norm@{x, y} < .15,
Red,
ColorData["BlueGreenYellow"][z]]]},
(* Define a color function that is red within some radius (here 0.15) and something else outside*)
Plot3D[(y^2/x), {x, -3, 3}, {y, -3, 3},
ColorFunction -> colf,
ColorFunctionScaling -> False,
PlotRange -> All,
PlotPoints -> 80]
]


and the result will not suffer from the problems that a thick point would have:

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