Why would an external USB disk drive be ignored by Windows 7, while it is perfectly working on other systems and USB itself is working properly on the target PC? It took me a while to find out. In the disk management console (under administration tools, computer management), I finally found a nice error message that was not displayed anywhere else: "The disk is offline because it has a signature collision with another disk that is online".
To fix that is actually quite easy, under a Windows 7 command prompt:
diskpart
Microsoft DiskPart will start. When it is ready, it will issue a "DISKPART>" prompt.
list disk
This will list all the disks that are currently mounted.
select disk x
Where x is the number of the disk from your "list disk" display. If one types "select disk 1", DiskPart will respond by telling you "Disk 1 is now the selected disk".
uniqueid disk
DiskPart will respond with the disk's signature, a series of hexadecimal digits.
uniqueid disk ID=NEW_ID
Where "NEW_ID" stands for the new identifier you want for the disk. And finally:
exit
It seems strange this problem can arise at all. However, the ID is not that long and is not a GUID. Cloned disks will get the same ID as the original, and I can imagine that manufacturers (HP in this case) don't care for changing this ID on every machine they build.