Following code should help.

function Button1_onclick() {
    var WebTreeView1 = ISGetObject("WebTreeView1");
    WebTreeView1.Nodes[0].Tag = "Tag is set on Button1_onclick event";
    document.getElementById("Text1").value = WebTreeView1.Nodes[0].Tag;
}

Please let me know if you have different scenario.

Actually , the main task was to change the Tag value on the currently selected node.  so somehing like this:

function Button1_onclick() {
    var WebTreeView1 = ISGetObject("WebTreeView1");

    var selectedNode = WebTreeView1.GetSelectedNode();

    selectedNode.Tag = "Tag is set on Button1_onclick event";
    document.getElementById("Text1").value = WebTreeView1.GetSelectedNode().Tag;
}

I realize that the changes made to the selected node variable are not sticking, but how can I make the modification to the currenlty selected node? 

Dasha, 

Did you mean that you need to access the Tag value from server-side during postback? If so, you can set the Tag value in client-side using the method below.

selectedNode.Set("Tag", "NewValue", true);

Using the above method, the value will be modified in client and also persisted to server-side during postback. Let me know if that works for you.

This worked! Thank you.