A good MCU which supports Arduino library but is not as expensive
Have a look at the ATtiny family of Atmel microcontrollers, and the several tutorials out there for using these MCUs with Arduino code, or a subset of the Arduino libraries. 


Begin with the well-known MIT post on the subject: "Programming an ATtiny with Arduino"
Download and study the Arduino-tiny library for ATtiny.
Follow this blog, at ToasterBotics, for another set of inputs on using an ATtiny with Arduino. 
See this Instructable, "Moving from Arduino to ATtiny and writing your code in pure AVR-C", for moving to the next level, beyond Arduino library limitations.


Alternatively, consider the very low cost Arduino clones available, for instance this Arduino Pro Mini clone for $3.56 including free shipping.



You will need an additional USB-to-TTL adapter for programming the boards, though.

If you want something inexpensive with an USB connector built in, look at the Arduino Nano clones, $6.88 with shipping:



These two options above will save the effort of designing and debugging a new PCB, if they already serve your purpose: They sell fully assembled and operational at prices you cannot hope to match without huge volumes. 

Finally, you can also consider moving away from the Atmel family entirely, as others suggest, if you are comfortable with reinvesting sunk effort in learning and coding for a new, cheaper platform.