Remember, first of all, that hosting and domain management are completely separate.
If your mother's hosting company allows for multiple domains on one account, then it will take you through a "wizard" of sorts to set up a new domain. On HostGator, for instance, there is an icon that you click to set up a new hosted domain (I believe the icon actually says "addon domain."
- It then asks for the new domain - we'll call it xyz.com
- It also asks you to name the subdirectory/subdomain - this is the subdirectory under which the new domain will reside on your mother's account. Let's say your mother's domain name is abc.com. You could create the subdirectory /xyz/ under her account. So technically there will be http://abc.com/xyz I think at the same time it creates a subdomain under your mother's account so you will also have http://xyz.abc.com
- It thirdly asks for a password for that addon domain.
Using an FTP tool like CuteFTP, you can then upload files using the new addon account name xyz.com and the password you created for the account. HOWEVER - you will not be able to do this until you have changed the nameservers as we discussed earlier! If you are itching to upload in the meantime, you can upload to the subdirectory under your mother's account - because that and the addon domain are one and the same.
In other words - there will be three ways to access this addon domain - via the actual domain name once the nameservers are changed - via the subdirectory (abc.com/xyz) - or via the subdomain (xyz.abc.com).
Again, I hope this made sense.
What you described above sounds like redirecting a domain, which is different from what I think you actually want to accomplish. If you simply redirect - then anyone typing in xyz.com will land on a page that shows abc.com/xyz (or xyz.abc.com) as the url. xyz.com will not show up in the URL area. You have to create an addon domain.