So right now I'm messing around with a 4060 binary counter for a timer circuit and am wondering about a switch that would be able to alternate positions each time I press it.

I have a rotary switch but it would be interesting if there's a pretty basic circuit I could make that would have the circuit change to a different position each time I press a momentary push button switch. If that makes sense? I'm having trouble figuring out what to google since I can't really find what it would be called.

Thanks

 

Have you thought about using a thumbwheel switch. There are several types available. BCD or HEX with diode isolation would allow a means of decoding values from your 4060.
http://www.digikey.com/product-search/en?FV=fff40011,fff80502,fffc0055