Sorry for my late reply.
Once you have created the project and added the websites, you cannot edit their names anymore. Also, you cannot add an alias name for them. You can only add an alias URL, in case you have multiple URLs which belong to the same company or domain.
A workaround solution would be to add the websites to the project with the synthetic name and then add the complete URL as an alias for them. For example, add a new website from Project - Websites and type in the Add Websites manually dialog, the name 'website'. Then, highlight the 'website' row, click on the Aliases button and add the URL as an alias: 'www.mysite.com'. However, this works for websites newly added to the project, not existing ones.
Let me know if this answers your question and if you need additional information.