Hi,
What is the Service Pack version of your SharePoint server 2003?
How many permissions does the problematic user have? Which SharePoint group does the user belong to?
Try to assign to another SharePoint administrator and check the effect. If you can assign the task to SharePoint administrator, it seems that the issue is related to permission.
Try to grant appropriate permission to the problematic user and check the effect.
In SharePoint server 2007, when you click column “assigned to”, you can choose from “all users” or “SharePoint group”, since I don’t have SharePoint server 2003 environment, but you can have a look at this on your SharePoint 2003.
For more information about control user rights and assign task, please refer to the following article:
Controlling User Rights and Assigning Tasks (Windows SharePoint Services 2.0)
http://technet.microsoft.com/en-us/library/cc287705.aspx
Managing Site Groups and Permissions (Windows SharePoint Services 2.0)
http://technet.microsoft.com/en-us/library/cc288440.aspx
Hope this helps.
Rock Wang
Rock Wang– MSFT