My friend's mother's death really annoyed me.
Why do you feel more annoyed than you do sad?
I hate the fact that her death affected me so much
It is entirely understandable to be affected by the death of your friend's mother. Maybe if you can focus on helping your friend, you will be able to cope more easily with all of these difficult feelings.
It's annoying that my friend is sad.
This just seems like a selfish reason to feel annoyed at a death. Your friend being said should motivate you to make them feel better, not be annoyed by them. Death is a terrible thing, and a great opportunity to be supportive of the people you care about.