You might be able to find an existing remote control with a suitable number of buttons, at a flea market, junk shop, or local rubbish dump.
Have a look inside and see what transmitter encoder chip it uses.
If you are really really lucky, you may then be able to buy the receiver decoder chip to go with it along with a suggested circuit diagram from the chip supplier.
The receiver section can be home built, but having a small robust, and nice looking commercial hand held remote makes it all so much nicer.