Open the project and do the following:
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Click on Details in the menu to the left
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Click on the pen to edit the project details

3. Go to the field Contact person incidents and search for the name of the person that should be the contact person. The person has to be added in the project and have accepted the invite in order to be selected
4. Click on Save

Contact person added:

NOTE! This can only be done by account owners or project admins in the project.