How to generate an unique computer id on Delphi?

Use the hard disk serial number or MAC address:

https://stackoverflow.com/questions/331106/how-to-get-serial-number-from-hard-disks (dead link)
Generating a unique machine id

Have a look at SysUtils.CreateGUID which creates a globally unique identifier. Syntax:

function CreateGUID(out Guid: TGUID): HResult; stdcall;

A small example taken from D2010's help:

{
This example demonstrates the usage of some GUID 
related routines along with the type itself.
}
procedure TForm2.FormCreate(Sender: TObject);
var
  MyGuid0, MyGuid1 : TGUID;

begin
  { Create a new GUID from the string representation. }
  MyGuid0 := StringToGUID('{00020400-0000-0000-C000-000000000046}');
  Memo1.Lines.Add('The GUID is: ' + GUIDToString(MyGuid0));

  {
  Accessing GUID's internal fields
  Using the Format function to obtain the same output as GUIDToString
  }
  Memo1.Lines.Add(Format('GUID using formatting is: ' +
       '{%0.8X-%0.4X-%0.4X-%0.2X%0.2X-%0.2X%0.2X%0.2X%0.2X%0.2X%0.2X}',
       [MyGuid0.D1, MyGuid0.D2, MyGuid0.D3,
       MyGuid0.D4[0], MyGuid0.D4[1], MyGuid0.D4[2], MyGuid0.D4[3],
       MyGuid0.D4[4], MyGuid0.D4[5], MyGuid0.D4[6], MyGuid0.D4[7]]));

  { Autogenerate a random GUID at run time. }
  if CreateGUID(MyGuid1) <> 0 then
     Memo1.Lines.Add('Creating GUID failed!')
  else
     Memo1.Lines.Add('The generated guid is: ' + GUIDToString(MyGuid1));

  { Generating second random GUID. }
  CreateGUID(MyGuid0);

  { Testing if two guids are equal. }
  if IsEqualGUID(MyGuid0, MyGuid1) then
     Memo1.Lines.Add('This cannot happen! CreateGUID guarantees that ' +
                     '2 randomly generated GUIDs cannot be equal!');
end;

HTH

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?