It depends what you mean by "make your own mobile phone".
A mobile phone module, like the Telit devices, or, as far as I can tell the simcom 300, is all the tricky bits in one box.
You need to add a SIM card holder, aerial, battery or power supply, microphone and earphone along with some sort of controller to tell the module to dial, answer and hang up. It is quite a lot of work, but you really don't need to handle rf or any of the tricky protocols involved in a mobile phone network.
The SIM just wires straight to the module. 4 wires + earth and all you have to do is get the wires to the right places.
Dialing is just sending "ATD 0123456789" followed by carriage return, so a bit of programming, but not too much is needed.
www.sparkfun.com made a rotary dial phone into a cell phone:-
SparkFun Electronics
I made a Trimphone with a rotary dial into a cellphone in a similar way.