In reply to edunn:
I'm sorry, but it always amuses me when I see the word 'cheap' in relation to Norway, which I believe is the most expensive country in the world. There's nothing remotely cheap about Norway, nor Rjukan for that matter. As said above, the cheapest place to stay - comparatively speaking - is probably the Old School Hostel in Bølgen, which is in downtown Rjukan, some distance from the centre so just popping out to the shops isn't easy without a car. It's contactable via FB only (as they no longer have a 'proper' website and clearly assume the whole world is now on FB.) It's reasonably priced but has limited space and books up quickly. Otherwise, places like Rjukan Hytteby offer nice but rather cramped accommodation with just about adequate self-catering facilities for which they will charge you an arm and a leg.
As for conditions in January, it'll be a bit risky as it's early(ish) season, but there again the same applies to everywhere else. Hopefully you'll be lucky and get really good conditions as Avinash Aujayeb did at the beginning of this year.
Alternatives? I've now been to Cogne a few times in mid January and it's been okay every time, but admittedly better some years than others, last year being the worst but we still climbed every day. If things are a bit slow in shaping up in the Cogne area by the beginning of January (which is unlikely), there's some really high stuff away from the Lillaz and Valnontey valleys that normally comes into condition in December (the start of the climbing season). This is located further west in Valsavarenche, Val di Rhêmes and Valgrisenche. Few Brits venture that far but it's well worth it. It goes without saying that Italy is also far far cheaper than Norway, even more so now with the current exchange rate.
I've also climbed extensively in the Embrun, Ceillac, Briancon, La Grave and Bourg-d'Oisans areas in mid January, but conditions have been far more miss than hit in recent years, so much so that we've stopped going there at that time of year.
Some food for thought if nothing else.