What are some applicable and acceptable forms of extracurricular activities to include on a Navy Officer package?

The whole point is to demonstrate initiative and involvement "above and beyond" the routine. Officers have a great many "un-official" obligations they may encounter...in addition to those more "official" collateral duties such as PAO, etc. Someone who already demonstrates the ability to manage a complex matrix of such responsibilities, is who they're looking for. Big Brothers/Big Sisters, Boy Scouts, Rotary, American Legion, various post outreaches, Church activity, etc...etc. They probably aren't looking for things like non-command related sports team participation (though I'm sure it's attractive in other ways), or if you run ultra-marathons.

Professional memberships related to your desired field or current field. Toastmasters is good as well

By this time in your Navy career you should know, The Navy loves community involvement. Quick ways to build up big projects are habitat for humanity, starting a homeless support program collecting materials and handing them out, tutoring programs at local schools. Remember while doing all of these you are there representing the Navy and what it can do for the non-military community, improving the lives of the local populace.