Both January and May are okay, but for May this year, you may encounter some difficulties to enter Tibet because of the 60th anniversary of the so-called "peaceful liberation of Lhasa Area". So coming in January is better.
And you will have a much better chance to see a clearer sky in January than May (though May is usually also okay for the weather.)
The best place to stay during January in the Everest region is the Ronguk Monastery Guesthouse, a bed is RMB 40 and a room (with nothing!) can cost up to RMB 200, there's no real way to book the rooms in advanced, but they are usually not full during January, unless you are unlucky to have another big tour group coming at the same time.
The monastery guesthouse will provide you enough blankets, but a sleeping bag will be helpful to keep you warmer.
If you don't want to stay at the Rongbuk Monastery, you can choose to go back to Tingri (no view of Everest) for the night. The Snowland Hotel (around RMB 180 - 220) is good for a night.