There aren't many reasons to visit Paris in January-- even February and March can be overcast and rainy-- but January can be quite cold, actually. There is only one reason I can think of to visit this outdoor city during the indoor season: the sales!!
Americans are generally well-versed in the Black Friday sales and the steep discounts the day after Christmas, but the French save it up for one HUGE sale during the month of January. In short, everything goes on sale-- clothes, shoes, handbags, haircuts, wine... everything (in French the word "soldes" means sale). During the course of the month, the discounts get bigger and bigger until they are about 70% off retail price. Then, February 1, the stores open with their new collections and the "soldes" signs are gone until next season (July).
One of the easiest ways to take advantage is to visit the large department stores near the Opera Garnier. Most of the significant shops in Paris have space in the large department stores-- everything from Gap and Zara to Sonya Rykiel and Ralph Lauren.
My favorite department store here is the Galeries Lafayette. The variety of collections are generally astounding (and cover a variety of sizes and price ranges) and the domed, Belle Epoque building itself is no less dramatic.
If you go, be sure to spend all your money at once. After you're done, you may visit the "detaxe" downstairs to get back the tax from your purchases. This can be a long process depending on the line, but it can also save a significant amount of money depending on what you spent. They will give you a form to take to the "detaxe" center at the airport upon your departure. A refund will then be sent to you.
Shopping in Paris is not only fun, but also a very cultural activity. Regardless of when you visit, I encourage you to spend some time exploring the shops and boutiques. Afterwards, I'm sure you, too, will be "soldes" on Paris!