My girlfriends daughter recently got an IPHONE for Christmas but the mom does not want her on Youtube (she blocks this site on her computer at home). Can I restrict access to youtube ?
No, you cannot. Please take a look at the many threads on content filtering for the iPhone and iPod touch (i.e. do a Google search), and you will find that no solution exists, nor probably will exist until Apple releases the SDK for it in February.
YouTube does a decent enough job at filtering their content. Personally, I would be more worried about the Safari browser on the iPhone than YouTube, since the browser can navigate to any non java/flash website...