You can easily assign a Keyboard shortcut to Family Browser following these steps.

  1. Click on the file menu pulldown in Revit and goto options.
  2. Click on User Interface tab onb the left hand side panel.
  3. Click on Keyboard shortcuts button.
  4. Filter for "Add-Ins Tab
  5. Scroll down until you find "Show/Hide"  "Add-Ins > Family Browser R3
  6. 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.