I checked the DNS settings and noticed they were not well configured. I discovered that he only set up two "A" records instead of the four "A" records needed for a Blogger custom domain and also noticed a server provided by the registrar - substituted for one of the four "A" records.
I reconfigured the domain name with the necessary settings and told him to wait while the local DNS server is refreshed. After few minutes, it started working perfectly and his .blogspot.com was successfully changed to his desired .com address.
NB: If you notice an error message telling you that "188.8.131.52" doesn't exist, add the remaining server addresses and refresh the page till it saved successfully.
Have you experienced similar problem while trying to set up a custom domain for your blog?