I'm not exactly sure what you're asking? It might help if you clarified why you need the "unique computer id".
Option 1
The bit:
"I was planning to store it in the registry and check it when the app starts, and if it changed update the registry."
gives me the impression you're trying to detect a different (newer) version of the app and/or identify different apps within a suite.
If this is the case, then simply pressing Ctrl+Shift+G
will generate a unique GUID for you and insert it at your current cursor position in code.
You may want to also look at the version information; perhaps auto-increment build numbers?
Option 2
If you're trying to in some way identify unique installations on different computers, take a look at the CreateGUID
function in plainth's answer.
Option 3
You mentioned that you "don't care if the id changes because of a hardware change or OS reinstallation". I'm not sure if this is an important fact like: you're trying to detect hardware changes? After all, you did say you don't care; but then why mention this at all?
Option 4
Something else I haven't considered?