Is there a more elegant way to draw attention to a new link rather than using animated GIFs?
If your application allows for it, I'd recommend branching out from standard text links. Your question doesn't specify the link use case, so I will give examples across some common uses. Consider the following alternatives:
If you have a new link within an existing UI element (such as a top menu), a simple badge gives a visual indication something is new. Make sure to get rid of it after the user has looked at it.
Advertising (especially on mobile) has increasingly used buttons to increase clicks. Buttons draw attention using color and space, as well as intuitively feeling more natural to click than text.
If you need to draw attention to a product, website feature, or another visual-based destination, encapsulate an image with a button.
If you have an important new announcement, site feature update, or other significant item that needs traffic, consider a banner. Make sure to allow the user to close it, and keep it hidden from future visits.
If you want your users to look at a new link first, think of adding contrast to the new link. Here are a few examples how you can do it -
More here Control the User’s Focus and Attention**
If your design is prioritised correctly then good or new content can 'bubble' to the top without the need for any flashy signs, graphics or pop-overs.
It's important to remember, the content may be new, however it may be completely uninteresting to the the reader or at least not what they are looking for. Bringing new and interesting content into the eyeline is good practise, forcing it down peoples throats - not so much.