You can easily assign a Keyboard shortcut to Family Browser following these steps.
- Click on the file menu pulldown in Revit and goto options.
- Click on User Interface tab onb the left hand side panel.
- Click on Keyboard shortcuts button.
- Filter for "Add-Ins Tab
- Scroll down until you find "Show/Hide" "Add-Ins > Family Browser R3
- Assign your keyboard shortcut and save.
You can also add the Family Browser "Show/Hide" icon to your Quick Access Bar, by right mouse clicking on the Family Browser icon in the "Add-Ins" tab and assign it to the QAB.