How can I replace the functionality of a button with an arduino?
You can add a parallel path with a transistor for current to flow from your 5V rail through the LED to ground. Effectively you're building an OR-gate, so the LED will be on if the button is pushed OR your micro-controller decides it wants the LED on.

Here's an example schematic using a N-channel MOSFET. The MOSFET is on when the IO pin is driven high, and it's off when the IO pin is driven low.