What part of the Arduino functionality would you like to keep?
At a minimum you need:
16 MHz crystral
two 22pF caps
100nF caps on VCC, AVCC pins
using analog? need 100nF cap on the Aref pin
10K reset pullup resistor
diode such as 1N4158 across the resistor (anode to reset pin, cathode to Vcc)
Header pins to connect FTDI Basic/FTDI cable, and 100nF cap to connect to Reset pin
header pins to connect to ICSP header for rebootloading or for installing bootloaderless sketches
place to connect 5V wallwart, or a battery, or a 5V regulator
Here's an example for a ATmega1284P type board, a '328P would use a smaller connector
This one has both FTDI and ICSP headers, but does not have the diode.
It has screw terminal for bring in 5V power from a wallwart or 4.5V from a triple-AA battery pack (I've used both)