How to create small engine with timer
You could use an Arduino microcontroller and wire it with a relay inside the button panel of the washing machine.  This might be a little neater than using a solenoid and wouldn't require you to take it off when you aren't using it.  You could have a dial (potentiometer) to adjust how long it will wait (1 hr, 2 hrs, 5 hrs, etc.) before momentarily clicking the relay.  A little indicator led could be used to show it is counting down.  An Arduino Uno might would be a bit overkill, but would allow plenty of customization (LED/LCD display, different methods of input) and also expansion (you could also control your dryer or other appliances). I would be happy to give you more information, I don't know how much experience (if any) you have in any of this.