Can someone show me a way to create a client-side variable in one module for another module on the same page to use?  I have tried playing with the ClientAPI (shown below), but so far it hasn't worked for me. The variable would be a simple string value. 

Here is what I tried during testing (following the example in the documentation):

IN CODE-BEHIND:

' In the page_load event
DotNetNuke.UI.Utilities.ClientAPI.RegisterClientVariable(Me.Page, "TestVar", "Sucess!!", True)

IN MARK-UP:

// at the bottom of the module mark-up
var oVar = dnn.getVar('TestVar');

[EDIT]

After that, I decided to try the other example of setting and retrieving the value on client-side, and when that line of JavaScript attempts to execute, the browser crashes.  Here is what I put in the mark-up just below the other JavaScript snippet from above:

dnn.setVar('newTest','SUCCESS!!!');
var oVar2 = dnngetVar('newTest');

[/EDIT]

You can always try setting a JavaScript variable to the value you want to be used in the other module..

In the code-behind of one module:
Response.Write("<script language=JavaScript>")
Response.Write("var prefix = dnn_ctr" & Module.Id.toString() & "_yourModuleName_ctl00_")
Response.Write("var myVariable = '" & strVariable & "';")
Response.Write("</script>")

On the client side of your other module, you can retrieve it like this:
<script language=JavaScript>
document.getElementById(prefix + 'txtHiddenField').value = strVariable;
</script>

<input id="txtHiddenField" name="txtPositionHold" type="hidden"  runat="server" />

Then you can access that value in the code behind of your second module if you reference txtHiddenField.  You might have to stick this at the top of your code behind:
Imports System.Web.UI.HtmlControls.HtmlInputHidden