Is it self powered (powered by the data cable) or does it use a mains electricity connection. Sometimes the USB it's plugged into just doesn't have enough power, I have a WD passport that refuses to work on some USB, especially ones that are carried over a wire, and not directly attached to the motherboard, like front access USB for example.
Lol. Ok. Can you give any information you know about the computer in question, does it have a model number?
Also, have you tried having the drive plugged in before you turn on the computer. Have you tried restarting your computer? (Sounds stupid, but sometimes needed to successfully integrate drive migration).