Get Website ID from Website Name

Websites are already loaded on startup, so there is no need for additional queries. You can and should use:

$websites = Mage::app()->getWebsites();

to receive them. Then to look up by name:

foreach ($websites as $id => $website) {
    if ($website->getName() === $name) {
        return $id;
    }
}

You need debug the collection by ->getSelect()->__toString()

for checking it give right value or not.

    $siteModelCollection = Mage::getResourceModel( 'core/website_collection' )->addFieldToFilter( 'name', $site );
// print the Query
    echo $Query = $siteModelCollection ->getSelect()->__toString();
    echo $siteId =$siteModelCollection->getFirstItem()->getId();

Update:

Use load():

for this case you need to call load() function of resource model of core/website for getting proper collection

Now you can get id of first store by using below code:

$siteModelCollection = Mage::getResourceModel('core/website_collection')->addFieldToFilter( 'name', $site)->load();
echo $Query = $siteModelCollection ->getSelect()->__toString();
echo $siteId =$siteModelCollection->getFirstItem()->getId();