Hi,
As Peter has said there are a ton of ideas to build a stopwatch. You can use a Real Time Clock but it will be cheaper if you use a mCU with a 12 MHz clock. This will give you a machine cycle of 1 microsecond. With timer 0 as 16bit timer you can load TH0 & TL0 with D8EF and timer0 will execute each 1/100 seconds.
In this site you can find ideas and code for the buttons and the LCD.
Good Luck
Alberto