Replacing simple physical switch by an arduino controlled digital one
From the information you have given in your comments, your original circuit should be fine. I am including it below for the benefit of others.



A few safety notes, since it does look like this switch is switching mains.


Always unplug the unit before working on it. Don't just switch it off at the wall - at some point you will forget and hurt yourself. If I am working on stuff that really scares me, I have an extension lead with a light plugged into it, and then plug the device I am working on into the extension lead. That way, if I only ever unplug and plug in the extension lead I have, there is a light on when the unit is energised. I used to work on x-ray power supplies, so it was a bit more critical to me then :)
Before touching any wires, use a multimeter to check for any residual voltage from electrolytic capacitors that remain charged or circuits that they are powering. This will normally be DC, but you might as well check on both DC and AC settings. The likelihood of there being any danger with this particular unit is very low, but learning good practise now will always be useful later.
Be neat and tidy with your wiring. If you nick insulation, cut the wire back and restrip. Don't have lots of exposed copper hanging out of screw terminals - strip back just the amount you need. Be sure to give all screw terminals a reasonable "tug test" to check that the wire is secure.
Be neat and tidy with your circuit boards! Think about where your relay module is going to go, make sure it is securely affixed in the final build. Make sure that the bottom side of the board isn't going to contact any other conductive material.