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.