Q:

How to add multiple copies of a widget from "available widgets"

I've found a widget that I need to add to more than one sidebar widget area.  However, apparently its been coded so that once you drag it from "available widgets" over to a sidebar widget, it disappears from the "available widgets" listing. 
I'd like to use it in more than one sidebar.  What determines this in the widget code? 

A:

Sounds like your widget written in the old (pre 2.8) style rather than using the widget class.  If that's the case, you need to add code in a few different places to handle multiple instances (it's not a simple option you can turn on or off). 
Check out this page in the Codex, and this article about creating a multiple instance widget linked from it. 
Depending on the complexity of the widget, it might be easier to just rewrite it using the Widgets class (examples are provided in that Codex article).  It handles multiple instances automagically.