If you wish to export a specific project hierarchy names, you can use the query field in the Custom List app setting to accomplish this. With type set to 'Project'. Enter a query such as:
((Name = "First Project") OR (Parent.Name = "First Project") OR (Parent.Parent.Name = "First Project"))
If your top level project is called 'First Project' and you have 2 child levels of child projects (3 total levels), this query will restrict your results to the 'First Project' hierarchy.
Please Note that the maximum hierarchy is 10 levels, so if you have additional levels that you wish to traverse, you can add additional syntax to pull those levels of projects as well.