Open Start Menu>notepad(right click)>Properties
Click on Shortcut Key and then press a shortcut key that is not used for anything else
Program a sketch for an Arduino that can use the Keyboard library that sends that shortcut key when the button is pressed.
This is a pretty simple solution that should work for opening most any application and any other action that can be triggered by a shortcut key. If you need to accomplish actions that aren't already controllable with a shortcut key you might want to check out EventGhost. If you need to control your computer from a greater distance than USB will easily allow then you can also send commands to EventGhost through a network using Ethernet or WiFi.