I've been in France a couple times in January. The average Minnesotan wouldn't consider those cold temps, but we are a tough folk.
When we were in Paris, it was 40 degrees F, and sunny. It was great for us.
IMHO, the bigger the city, the better it can be handled in less than ideal weather. Lyon is a largish city. I'd go. They have a large population so I would think most places, other than outdoor venues would be open.
In addition to sticking to larger cities for winter travel, you can think about going to a place for "true winter". A place that doesn't have transitional weather, so like the mountains. You might think of a couple days in Chamonix.