Hierarchical data using closure table, how to ensure that a column is unique within the parent node?
I'm assuming (from the fact you considered adding a parentId column) that each Node has a single parent.
Two things to do:

Add parent column to Node, with a self-referencing foreign key to the nodeID column.  This ensures that each Node links to a single parent Node.  This should be null for the root node.
Get rid of NodeClosure table.  Seriously, you don't need it.

So your design becomes simply:
Node

nodeID (int not null, autoincrement, primary key)
parentID (int null, foreign key to Node(nodeID))
name (text not null)

Now you can indeed enforce unique names among siblings by adding a unique constraint on parent + name.
You can derive the ancestor/descendent relationship and level that was previously in NodeClosure using queries against the Node table.  Syntax for this varies depending on database.