Duplicating GameObject with variations of the same functionality.

I've written a script that extends the MonoBehavior class. It essentially turns a game object into a "beacon", which can be selected, moved towards, etc. Each beacon should have unique properties like a name and other such information.

I'm new to Unity and unsure of how to do this, but how would one reuse that script and still give each object unique properties? In other object-oriented languages, we create instances of a class with a constructor, which takes care of my problem because the unique values can be specified there, but I'm not sure how to do the equivalent in Unity (C#) since I'm just creating a new game object, placing it on the map, and attaching my script to it.

Thanks!