Making an object heavier on one side to afect the fall of an object with ridgid body (active) applied to it

Something with an off-center center of mass doesn't fall any differently in real life (disregarding air resistance, which is not simulated by Blender's RB physics). It won't rotate as it falls unless it has some rotational inertia from some other cause; all parts of the object will experience the same acceleration due to gravity.

Rigid body physics places the entire center of mass of an object at the origin of that object. (That means that properties like moment of inertia are not simulated.) You can create an off-center center of mass simply by setting the origin of the object. If you do this, be sure to use mesh or convex hull collision types, as other types of collision bodies are centered on the origin and won't have the proper behavior for an object without a centered mass.

There are two ways I know that I think may work for this: 1) Add a small mesh inside of the object you want to make fall, and set the weight to a higher value. Then position it where you want the object to be heavier, and join the two meshes. 2) Animate the object while it is falling.