When I start to enter an e-mail address in the TO field on an e-mail, I sometimes get names I don't have in my Address Book. The names are ones to whom I had sent one e-mail in the past or who sent to me.

How can I delete these unwanted, hidden names?

In Outlook 97, right?
Those names are stored in the nickname cache. I'm not sure where this would be found with Office 97 / Windows 7., but back in Office 97 it was a .nick file so you could close Outlook then search for *.nick and see if you find it. It's probably named YourUsername.nick.

If you find it, rename it to oldnick.old
When Outlook doesn't find the .nick file, it will make a new one.
of course, you wil lose all your nicks.

Easier though is to delete them as they pop up in Outlook. You should be able to select one using arrow keys, then press the delete key.

Microsoft also released a NickName.exe utility to manage the list while preserving the profile.

See: http://support.microsoft.com/kb/164598/EN-US