Can WebScriber collect the names of subscribers?

Unfortunately WebScriber doesn't support collecting additional fields when a user subscribes to a mailing list. WebScriber only acts as a web interface to EZMLM, and still uses EZMLM for storing email the addresses, so there's no facility for storing additional data with the addresses.

If you need this functionality and don't have to use EZMLM / don't mind using a hosted solution, you might take a look at MailChimp instead. They can collect names and many other demographics about your subscribers (such as location and even Gravatars photos). At the time of writing they offer a free plan for up to 2000 subscribers.