Rotating shape around defined point instead of centroid in QGIS?

To rotate feature(s) in QGIS you need to be in editing mode, select one or more features then use Rotate feature tool: , also found in edit menu.

While holding ctrl left click to move rotation centroid. I couldnt find a way to snap roatation centroid to points but you can zoom in very far to get sub milimeter accuracy. When satisfied with location just left click to start rotating (free or by typing in angle).

For certain similar task also plugin Vector Bender can be useful, it "Does to vectors what georefencers does to raster"

In QGIS 3 you can use the native rotate tool which can be found in the toolbox. Here you can specify an anchor point by clicking on the map in order to rotate around it at a specified degree clockwise.