Okay after more investigation I have figured out how to set this attribute.
The System Category is the built-in category that all livelink objects inherit. It specifies things like the Name, Creator, Create Date, etc. You can extend this category and add your own attributes. In this case that's what the "Title" attribute was.
To set these extended system attributes it's a bit different than setting other category attribute values. It's also different than setting standard object data attributes.
To do it you use the GetObjectAttributes and SetObjectAttributes functions.
Finding this in the API documentation is like finding a needle in a hay stack. One sentence mentions it, I easily overlooked it:
"This function sets the attribute information for custom <System> attributes."
If you refer to the documentation for those two functions it will point you in the right direction on how to set custom system attributes, but here is some simple sample code to set a custom attribute "Title" on a particular folder:
Dim attribs As LLValue = (New LLValue).setAssocNotSet()
status = m_Document.GetObjectAttributes(m_WorkspaceVolID, llFolderID, attribs)
If (status <> 0) Then Throw New Exception("Error!")
// The title attribute is the first rec value in the attributes
attribs.setString(0, "Value", "Test Title")
status = m_Document.SetObjectAttributes(m_WorkspaceVolID, llFolderID, attribs)
Even though I figured it out, thanks for the help guys!