Manipulating row data with Javascript

It sounds as though at some point, you have a reference to the link the user clicked (either because you have a click handler on it or because you're using event delegation and finding it after a click on the table). Starting with a reference to that link, you can use Prototype's DOM traversal stuff to find the second table cell:

Edit Based on your response to rahul, I would change your link onclick to:

onclick="handleLinkClick(this); return false;"

...and this would be handleLinkClick:

function handleLinkClick(link) {

    // Original code (mostly unchanged)
    Element.hide('currentdiv');
    new Ajax.Updater('someajax', 'ajax.html', {
        asynchronous:true,
        evalScripts:true,
        onComplete: function(request) {
            new Effect.Appear("newdiv",{});
            window.scrollTo(0,0);

            // New code starts here

            // Extend the link element
            link = $(link);

            // Find the row
            var row = link.up('tr');

            // Find the second column
            var cell = row.down('td').next('td');

            // Change the cell's "class" and "value" -- I've had to guess a bit at
            // what you want to do here
            if (cell.hasClassName("true")) {
                cell.removeClassName("true").addClassName("false");
                cell.update("No");
            }
            else {
                cell.removeClassName("false").addClassName("true");
                cell.update("Yes");
            }

            // End of new code
        },
        parameters:'authenticity_token=' + encodeURIComponent('SYWsdBTWlz17u9HmPXA2R9WmBfZn67g/IAMGyhHEwXw=')
    });

}

That uses Element#up, Element#next, Element#hasClassName, Element#addClassName, Element#removeClassName, and Element#update; docs for them here.

Optional things to consider:

• The above is fragile in that if you change the location of that cell (make it the third column rather than the second), it fails. You might use a marker class to find it.
• Rather than an onclick attribute, you could use Element#observe.
• You can use event delegation to have just one handler on the table, rather than a handler on each link.

But the above should work.