Yes, it's possible. Since Artboards are just hacked Groups (which are just hacked layers) what you needed was a script to get data from the active layers — to be able to select them one by one and rename. Here's how you do it (in this example the function will rename all the selected artboards to "my name index"):
var newName = "my name";
var ids = getLayersIDs(); // getting ids of selected layers
//for each id in the list
for (var i = 0; i < ids.length; i++)
{
// select the layer first (well, artboard in this case)
selectById(ids[i]);
//rename it to "my name 1", "my name 2", etc
activeDocument.activeLayer.name = newName + " " + (i + 1);
}
// this will get IDs of selected layers/groups/artboards
function getLayersIDs()
{
var lyrs = [];
var lyr;
var ref = new ActionReference();
var desc;
var tempIndex;
var ref2;
ref.putProperty(stringIDToTypeID("property"), stringIDToTypeID("targetLayers"));
ref.putEnumerated(charIDToTypeID('Dcmn'), charIDToTypeID('Ordn'), charIDToTypeID('Trgt'));
var targetLayers = executeActionGet(ref).getList(stringIDToTypeID("targetLayers"));
for (var i = 0; i < targetLayers.count; i++)
{
tempIndex = 0;
ref2 = new ActionReference();
try
{
activeDocument.backgroundLayer;
ref2.putIndex(charIDToTypeID('Lyr '), targetLayers.getReference(i).getIndex());
try
{
desc = executeActionGet(ref2);
tempIndex = desc.getInteger(stringIDToTypeID("itemIndex")) - 1;
}
catch (e)
{
tempIndex = 0;
}
}
catch (o)
{
ref2.putIndex(charIDToTypeID('Lyr '), targetLayers.getReference(i).getIndex() + 1);
desc = executeActionGet(ref2);
tempIndex = desc.getInteger(stringIDToTypeID("itemIndex"));
}
lyrs.push(desc.getInteger(stringIDToTypeID("layerID")));
}
return lyrs;
};
// this will select a layer by ID
function selectById(id)
{
var desc1 = new ActionDescriptor();
var ref1 = new ActionReference();
ref1.putIdentifier(charIDToTypeID('Lyr '), id);
desc1.putReference(charIDToTypeID('null'), ref1);
executeAction(charIDToTypeID('slct'), desc1, DialogModes.NO);
};