Language and identity
Learning a language that was formerly spoken in your family can be a way to enhance your identity as a member of a particular ethnic or cultural group. It can also help you understand the culture and thought processes of your ancestors.
Alternatively you might consider learning a minority language spoken in the area where you live and/or work. This can help you become part of the community of people who speak that language and might open all sorts of interesting opportunities.
In either case you could try try to learn as much of the language as possible, or just some basic phrases to throw into your conversations when you get the opportunity. Even if you don’t intend to learn more than the basics, you might find yourself becoming more and more interested in the language and learning much more of it.
If you learn a minority language, for example Luxembourgish in Luxembourg, Navajo in New Mexico or Basque in Spain, you are demonstrating your respect and conmittment to the local language and culture, which people are likely to appreciate. They may also treat you differently (in a good way) as a result. If the language in question is endangered, as many minority languages are, by learning it and using it you are playing a part in keeping it alive.