hobby noun , plural hobbies
: a pursuit outside one's regular occupation engaged in especially for relaxation
Yup, I think gaming fits the definition for the most part.
Of course, it shouldn't get in the way of other life priorities, such as family, school & work.
Also there are both positives and negatives. On the plus side, it can help keep the mind active, engage in new learning, promote problem solving, and in some cases, promote social interaction (just look at the forums). On the negative side, it can be addictive, certain games may encourage poor behavior for those prone to it, and it's possible the light emitted by screens can inhibit sleep if played late at night.
As with just about all hobbies, there are pros and cons, and it's up to parents to do what they think is best for their kids. For instance, we never allowed any screen time for our kids until they were two, and then at that point, it was limited to under an hour a day, and the programming had to be slow paced for younger children. Now our kids are full of imagination and rarely beg for certain toys, because they almost never saw commercials, lol.
I hope all goes well with your mom. I encourage you to do your best to show respect when discussing your point of view, as well as for whatever decision she makes. My guess is she's doing what she believes is best for you in the long run. No parent is perfect, but most of us try to do what's best for our kids!