This webpart is provided with the feature "BasicWebParts" (ID: 00BFEA71-1C5E-4A24-B310-BA51C3EB7A57).
It's a hidden feature, activated by the out-of-the-box site template. See
C:\Program Files\Common Files\Microsoft Shared\Web Server Extensions\14\TEMPLATE\SiteTemplates\MPS\XML\onet.xml
for example.
In your case, are use using a custom site definition ? If yes, add this feature to your onet.xml (you should do the same for all features of a blank site, aka STS#1).
If not, you can try to add it using a simple :
Enable-SPFeature -identity 00BFEA71-1C5E-4A24-B310-BA51C3EB7A57 -Url http://localhost
But you should really wonder why this feature is not activated... maybe your site is messed up and creating a new one can be a good idea (if you can...).
Get-SPFeature -Url http://localhost
. Do the same on the target box, diff the activated features, and try to activate the feature until the webpart is present.