Making watch or stopwatch isn’t a rocket science. All you need is to set up microcontroller timer and capture interrupts to update time. rwb has made pretty simple stopwatch which uses 8 digit seven segment LED display. Display is driven using MAX719 chip that connects to microcontroller via SPI interface.
He made great work on documenting project. He takes through all steps of process starting with hardware. He speaks on how to connect display and even how to expand to 16 digits and more using cascading of driver chips. Then he goes in to details on how to set up Atmega328P timer, generate interrupt and update time. Here you will find good info on running external interrupts that are used to detect button click. So even if you don’t intend building stopwatch it is worth looking for good tutorial material with great illustrations.