How to Implement the web part in such a way that it allows user to select name through dropdown/list selector instead of entering it manually in the edit mode.

The dropdown should populate all the list available on the current site.

It sounds like you need to create a custom toolpart for your web part. Please read this article for more information about how to do this. You would place your logic in the CreateChildControls() function of the custom toolpart class to populate the lists from the current site.

Hello dannyjessee,

I want to create a dropdownlist and in that ddl I want to show all my sharepoint list.

How can I do that?

Which webpart should I use?

Is there any article on that?