Once you've gone to the trouble of ridding your abode of unsolicated advertising, and you've started making sure your purchases do not introduce more advertising into your space, your friends and family continue to give you gifts that you wouldn't want in your house.
At the same time, you know of someone else who would appreciate the gift, and might not have as strong feelings about the advertising content as you do. You might as well unload the item on them at the next available gift-exchanging event. For that matter, why wait ? Just go ahead and make that other person happy.
In general, gifts received should go through the same set of preventive methods as any incoming product into your house. See the comments in the buying for unbranding method. In the case of the unwanted gift, the recycling methods are also of particular interest.
You may want to look into exchanging the gift for a model with less advertising. This is a good route to go if the givers are the type who would actually notice if the gift wasn't on display the next time they visit.
Another approach, which I'm not gutsy enough to do myself, but which others might try is refusing the gift. The only time I could see myself risking a friendship by refusing gifts would be if I had kids and someone was trying to give them war toys.
Of course, it's always possibile to donate the gift to charity, especially if the giver is never going to remember the gift in question.
Pick a person who would appreciate receiving the gift, and give it to them at the next available gift-giving occasion. You might want to make sure the new receiver and the original giver do not know one another, lest you end up in your own personal Seinfeld episode.
No special tools for this one. If you're really thifty, you may want to hang on to the original wrapping paper used with the original gift.
No special skills needed.
Very little effort for a modest payback. This is mainly a passive method so there's no feeling of accomplishment once the method is complete, plus you're left wondering if the receiver and original giver will ever figure out you regifted.