The gift shops at the Eiffel Tower are managed by the Société d'Exploitation de la Tour Eiffel…read more(SETE). There are several gifts shops, some on the ground level and then others higher up. Lots of cool Eiffel Tower things. I bought a baseball cap and a coffee mug.
From the SETE website - what they sell: "Eiffel Tower miniatures and key rings, post cards, medals as well as stationery, clothing, tableware... These items, some of which cannot be found elsewhere, are suited to all ages and budgets." If you are looking to find Eiffel Tower socks, however, well, I had no luck....
You can click on the Yelp link to their website and purchase from there if you forgot to get something when on site.
Interesting fact - according to Wikipedia the SETE "now considers any illumination of the tower to be under copyright. As a result, it is no longer legal to publish contemporary photographs of the tower at night without permission in France and some other countries."
Another interesting fact - recently (December 2014) gift shop workers went on strike to demand a 300 Euro salary raise (a month?). There are apparently about 50 employees. There are about 7 million visitors to the Eiffel Tower a year.