Unfortunately, the license for Lion or Mountain Lion purchased at the app store is not transferable. That is why it is so inexpensive.
You will need to buy your own. In order to do that at this point, you will need to purchase Snow Leopard first, available from Apple for $20 or, if it came with SL, ask them for replacements install disks by giving them the serial number (you have to call them and they will send an install DVD). You can then boot from the install disk and reinstall the system. After that, you can simply use Snow Leopard or pay to get Lion or Mountain Lion.