Introduction
Mathematica 9 has introduced a new predictive interface. For each output, it suggest possible actions the user might want to take.

How can this interface be extended so that it will suggest additional, possibly user-defined, actions?
The starting point
@Rojo mentioned in chat that he discovered that much of the functionality is contained in the Predictions` and PredictiveInterface` contexts. Many of the symbols have usage messages (but make sure that you trigger loading them by using the predictive interface at least once).
The Predictions`PredictionRule symbol seems to contain the rules on which the predictions are based, as downvalues (see its usage message).
The functionality seems to be defined in files found in the $InstallationDirectory/AddOns/Applications/PredictiveInterface/ directory. The relevant part is in .mx files (not directly readable) and comments in the .m files suggest that these may be updated automatically.
I posted this question as a starting point for people who want to figure out how this works, and to facilitate collaboration.