Lengthening button press signal
The astable multivibrator will send many electrical pulses once it is turned on,regardless of the fact that the button is pressed or not.Or,if you wire it so it activate on button press(and not by another source) it will send impulses with a time delay continously ,only while the button stays pressed.It's not useful.  

If you wire a capacitor to the button,you won't obtain much,but you can get the task done if you build a monostable multivibrator with one,from discrete components,that is:http://www.electronics-tutorials.ws/waveforms/monostable.html. This kind of circuit will emit a continous pulse per press and will continue emitting until a capacitor discharges,even if it doesn't get an input signal anymore.Note that if you press the button again,before the output switches off ,the multivibrator will be triggered again and the output will constantly send the pulse,as if you kept the button pressed this whole time.This is what you need.You can also use a 555 chip , as transistor stated:http://www.electronics-tutorials.ws/waveforms/555_timer.html. There are many other examples out there,on the web.

Here is a simple one:







simulate this circuit &ndash; Schematic created using CircuitLab

When the button is pressed,the capacitor is charged instantly and when you let go,the capacitor will discharge through R2 and keep the transistor running as the light shines.

Note that, when you're rewiring ,you're going to need a transistor(BJT NPN) to act as another switch.Cut the button off ,place it in one of the circuits I mentioned ,connect the cut wire which carries the current from the source of the game to the collector of the transistor,the other to the emitter and the output of the multivibrator to the base.I don't really know how your game's setup works,so this paragraph might be inaccurate.If you want to know how to actually do it,post another question and give details about your setup.