"Abort retry or fail?" is the message that comes up when the system has detected that it could not read data from a disk. Are these floppy disks? Whatever, the message means "damaged disk". When you say that you have worked with HP customer care on this, what did you mean exactly? I would imagine that a system that came with Windows ME must be at least 5 or 6 years old, if not more, and I would not be surprised if HP are not interested. If the machine in question is not the one that the disks came with, I would be even less surprised.
Your cousin will need to examine his or her options. These may include purchasing a Windows ME install CD on Ebay (or getting one in some other way that we cannot discuss here.)