Most of these recommendations are not open-source, but they are free and add a lot of useful functionality to ArcMap.
Geospatial Modeling Environment - Written for ArcView (works with all licenses though), adds some functionality that is available in ArcEditor and ArcInfo. Not open-source (uses R, which is open-source though), but free.
ET Tools (GeoTools, GeoWizards, Surface) - Lots of great tools. GeoWizards adds some of the 3D Analysis functionality. Again, not open-source, but lots of free tools available before you have to pay.
ArcScripts - Discontinued, but still an extensive archive of scripts contributed by users. All are free. Most of the scripts are pre version 10, but since you have access to the source code, you could port them over.
ArcGIS Code Gallery - New version of ArcScripts