An SSR is not a good choice for your application. The common use of an SSR is for switching on and off an AC load. If the data sheet mentions SCR or Triac, then it is meant for only switching AC and will not function for controlling DC. So keep this in mind when looking for components: if it says "24-220 VAC" on the data sheet that does not mean it will work for DC.
A second common use of SSR and microcontrollers is the opto-coupled input type. That input method provides a safe scheme for a low voltage low current control signal to switch a large load.
For a button on a phone, you are switching a near zero current DC load. You should be able to control that with a transistor, but that would require some reverse engineering of how the phone keypads where being scanned. You would need to trace out what the voltage was, how much voltage drop could be accommodated.
A certain easy method is to use a small reed relay. Parts are available that have a coil that only draws ~10 mA so can be driven directly from an output pin. The relays are packaged in a DIP so easy to prototype on a breadboard. Many list the coil resistance and voltage for specs, for example a 5V and 500 ohm resistance coil would draw 10 mA. You will find many for 1 - 2 $US.
When working with a relay, you need to take care of the back EMF (aka flyback voltage) that occurs when you turn off. That is handled with a single diode which you will see in every schematic for driving a relay. For a beginner, an alternate/sure success route is to use a driver chip such as a ULN2003 that will drive inductive loads such as small relays.