Renaming a Symbolic Link is done from the properties view of the page containing the link.  Each link in the list will have a '''Rename''' link that will initiate the renaming.  The link contains a URL of the form:
!***********************************************< HIDDEN
!define CODE {!-<pre><font size=-1>-!}
!define code {!-</font></pre>-!}
!define B    {!-<b>-!}
!define b    {!-</b>-!}
!define VAR  {!-<font color=blue><em>-!}
!define var  {!-</em></font>-!}
********************************************************!
${CODE}${B}!-http://host/PathToPage?responder=symlink&rename=-!${b}${VAR}nameOfLink${var}${B}&newname=${b}${VAR}newNameForLink${var}${code}

But first we need to create a link to rename.
!include CreateRelativeSymbolicLink

When we look at the properties page we'll see the rename link.
!|Response Requester.|
|uri   |status?|
|LinkingPage?properties|200|
!|Response Examiner.|
|type  |pattern|matches?|contents?|
|contents|SymLink|true||
|contents|<a href="javascript:symbolicLinkRename\(.*\);">Rename</a>|true||

Now click the '''Rename''' link.
!|Response Requester.|
|uri   |status?|
|LinkingPage?responder=symlink&rename=SymLink&newname=NewLink|303|

This should change the name of the symbolic link and redirect back to the properties view.
!|Response Examiner.|
|type  |pattern|matches?|contents?|
|contents|Location: /LinkingPage\?properties|true||

When we look at the properties page again, the symbolic link will have a different name.
!|Response Requester.|
|uri   |status?|
|LinkingPage?properties|200|
!|Response Examiner.|
|type  |pattern|matches?|
|contents|NewLink|true|
|contents|SymLink|false|