We all have met people like this—and you are right: Someone who constantly complains or talks about their problems all the time soon wears out their welcome. On one occasion, Jesus had to tell His disciples to "Stop grumbling among yourselves" (John 6:43).
Have you ever asked yourself why your cousin is like this? I don't know the answer, of course, but sometimes a person who constantly talks about their problems is actually just lonely and looking for attention. Do you ever talk over the phone with your cousin, or spend time with her except on holidays? Ask God to help you be a friend to her and show concern for her—even if it isn't easy at first. The Bible says, "A friend loves at all times" (Proverbs 17:17).
Then help your cousin realize that her constant complaining is actually hurting her by alienating her from others. She may not even realize how negative she sounds, and how others react to her grumbling. Don't rebuke her harshly, but let her know you care about her, and you want to help her.
The most important thing you can do, however, is to point her to Jesus Christ. Urge her to open her heart and life to Him, and then to put her problems into His hands. The Bible says, "Cast your cares on the Lord and he will sustain you" (Psalm 55:22).