I've added a few improvements and additions:

• Noise octaves.
• Random seed.
• Different noise calculation for each axis.
• Paintable weights (off the bat).
• Better UI controls.
• Removed the noise handling control (not much use in practice).
• It's now faster (I'm setting all the point positions at once, not one at a time).

Usage:

Download the .py file from the above link, and put it into your Maya plug-ins directory. Load the plug-in via the [Window > Settings/ Preferences > Plug-in Manager] (you might need to click refresh).

Select an object, and in the Python script editor, type:

import maya.cmds
maya.cmds.deformer( type='basicPerlinDeformer' )

Or, select an object, and in the Mel script editor, type:

deformer -type basicPerlinDeformer;

The position of the noise is controlled with a 3d texture placement node, move that to move the noise. You can paint the noise weights by selecting the object and in choosing [Modify > Paint Attributes Tool].