Dataiku DSS users can be assigned one of the following 3 profiles (by default): READER, DATA ANALYST, DATA SCIENTIST.
READER and DATA ANLAYST have some restricted actions (that you cannot modify). READER for example can not edit any project (even if they are in groups that allow project modifications).
In addition to profiles, users belongs to groups. From the documentation: "On each project, you can configure an arbitrary number of groups who have access to this project. Adding permissions to projects is done in the Project Settings, in the Security section."
So make sure that user has the DATA SCIENTIST (or DATA ANALYST) profile and belongs to a group that has the "Write project content" permission on the project.