Lake Como Area in Late January

Yes, most everything is closed for winter in Varenna. But if you get a hotel reservation, then that part's confirmed, and they can probably advise you of which few restaurants are open, or explore restaurant web links at http://www.discovervarenna.com/eating-out (and take promised open dates with a grain of salt). See ferry schedules at http://www.navigazionelaghi.it/eng/c_areadown.asp; I also saw notices that Cadennabia and Bellagio ferry ticket offices are closed for the season.

A larger town with less exclusively-tourist focus will have more going on in winter, and perhaps more indoor sightseeing like museums. I'd consider Venice or Siena over Milan, each about 3 hours away by train, not without tourists, but with plenty of old-world atmosphere and corners to explore.

I'd encourage you to check around the net and look at "when to go" sections of some guidebooks.
When you do, I think you'll find January is not a good time to visit the Lake Como Area, because of the weather and closures of lodging and other tourist services.
It's a great place to go at other times of the year.
Happy Travels!

I wouldn't go to Varenna or anywhere in the lake district in January. My husband and I were there in mid July. The weather still wasn't great yet. And there is practically nothing to do there, rain or shine, except enjoy the views and wander the alleys. Throw in the rain you'll likely have in January, as we had in July, and that will make for a very boring trip indeed.

What about going to Switzerland or Slovenia? Neither are very far from Milan and both have more interesting sights.

i want to disagree with Jordan, we were in Varenna AND Slovenia (lake bled) in July, and I adored Varenna. I think it depends on how you like to fill your time when he is writing.......I can recommend some apt, some of which are just a few steps from the small grocery store. I can also recommend some fabulous restaurants.....don't know about their winter hours, tho. assuming the ferry is running, you could do an easy day trip to Bellagio if you need more options.

I think Laura has made an excellent suggestion with Venezia. It really isn't a "city" like Milan or even Siena. It is quiet, without vehicular traffic, and uncrowded at this time of year, but everything is open. You can day trip to the islands (on a clear cold day), Padova, Vicenza. Rates are excellent: You can get an apartment overlooking a canal without breaking the bank.

January is on average the coldest month of the year, although not necessarily the wettest.
July is on average the hottest and the driest month of the year (in spite of the fact that this past year it was an unusually cool and wet summer in Italy)
As a rule of thumb, to estimate what the weather will be like, you should not rely on the anectodal evidence of people who post comments here based on one time experience. Rather you should look at statistical series published on many weather websites, such as Weather Underground (www.wunderground.com).
In January the weather is likely to be cold and damp. Cities offer the most indoor activities. The lakes are better visited during the nicer months.
If you don't like Milan (I don't either) you can base yourselves in another city. Venice, Florence, Bologna, Verona, etc. are all within a couple of hours away from Milan by train, and all would be excellent bases for the surrounding areas as well.