How to use:

• Select some vertices in the Weight Paint or Edit mode.
• Activate the tool with menu Weights > Adjust Vertex Weights (in Weight Paint mode) or from a hotkey that you set up (explained in a section below), or by pressing Space/F3 and searching for the name "Adjust Vertex Weights".
• Move the mouse up and down to offset the weights of selected vertices. If they're skinned you'll see them moving around as their weights are changed. The weights from locked groups will stay the same.
• Hold Shift for finer changes, press Enter to accept the changes or press Esc to cancel.

Note: when in Weight Paint mode, make sure that the Vertex Selection Mask button is pressed so that you can see what vertices that you've selected and will be adjusted. As a safety precaution, the tool won't start unless this is on.  

Recommended Workflow:

• Pose your mesh to find out where the skinning needs improvements.
• Select the vertices that you want to adjust.
• Lock all vertex groups (Vertex Group panel > dropdown menu > Lock All).
• Unlock only the vertex groups affecting the selected vertices*.
• Call Adjust Vertex Weights and change the weights to your liking.

* This step can be done a lot more easily by using a (free) add-on of mine, Popup Vertex Group List.

Keymap

You can manually add a keymap to this tool by following the Blender manual on this topic. The operator name is paint.adjust_vertex_weights to be added to the 3D View > Weight Paint.
If you want to use it in mesh Edit mode, you can also add it to the 3D View > Mesh category.

You can also add that operator name to pie menus and custom panels as you wish.

Removal

To delete this add-on, find it in the Paint or User categories of your Add-on preferences, expand its info box and press the "Remove" button.