I have a polygon in QGIS 2.14 and want to generate a specified number of coordinate points within that polygon, with a minimum distance apart. The fTools Random Points tool doesn't give this option, and neither does the "Point sampling" analysis tool in the Point Sampling plugin (or maybe I don't know how to use it properly). Is there a way to do this in QGIS? I know it's quite straightforward in the Random Points tool in ArcGIS.
You can use Random points inside polygons (fixed) from Processing Toolbox. If you don't see Processing Toolbox panel, activate it from menu
Processing --> Toolbox. For distances in meters you have to use projected CRS (e.g some UTM) for more information about CRS see QGIS Doucumentation
Point sampling tool that you mentioned, is good for retrieving values from raster layers.