A new hard drive needs to have the Apple TV operation system installed on it. Search on the Internet for an Apple TV 1st Generation restore image and follow the instructions. It involves copying it to a USB memory stick in a special way, inserting it into the Apple TV, and booting the Apple TV while holding down the Menu and + buttons on the remote control.
The Apple TV 1st Generation connects to a hard drive with the old Parallel ATA (PATA) interface, not the newer Serial ATA (SATA) interface. I’ve read that it can use a Compact Flash (CF) card with a CF-to-PATA adapter (and it runs much cooler than having a spinning hard disk), but I haven’t had any success with the adapters I’ve tried. Make sure it is an adapter to the 44-pin variant of PATA that is used for laptop hard drive — not the more common 40-pin PATA interface.
One of the things stored is the device name. When you rename the Magic Trackpad (System Preferences > Bluetooth), that name is saved on the trackpad. It comes up with the same name if you pair it with a different computer: a nice touch!