Always start with activities you enjoy (vs. ones that someone else enjoyed that you hope will boost your resume). If you love an activity, you'll likely be more inspired to continue it and to move up/excel/lead in it.
As I posted recently on another thread, the key for homeschoolers is twofold. First, show you can work and play well with others outside of your home and outside of the (usually) small group with whom you study. There are many organizations out there who would love to have someone join. Whether your interest is theater, journalism, art, music, animals, helping the disabled or elderly, etc., look for activities that cause you to engage with others. Then, once you're involved, look for ways to lead in those organizations.
Second, participate in sports. Team sports (see above) are good. Try not to make a sport such as martial arts or training by yourself for a marathon your ONLY sport, as these can be perceived (accurately or not) as "loner" sports. In any event, you want to demonstrate you can handle the physical rigors of USNA. So look for leagues and other activities that don't require you to attend a particular school. This could also add to your "team building" resume.
Again, the fact that someone else did a particular activity and was appointed doesn't mean that, if you do the same activity, you will have success. The concern USNA has about homeschoolers (whether or not a legitimate concern) is that they are loners who don't know how to / haven't had to be involved in school, clubs or sports with other young people and also they don't have the requisite level of physical fitness . As noted above, the key therefor is to show you are physically active and can work and play well with others, especially others outside your "comfort zone." You should do that in a way that brings you joy and fulfillment.