I can endorse Edinburgh just before or just after Christmas. I first came here in January, fell in love with the place instantly, so much so I moved here.
I would give Hogmanay itself a miss though, the street party is overpriced and miserable if it rains.
Apart from the bad winters of 2009/2010, it isn't that cold, if you're used to a winner in the rest of the UK. (I am constantly baffled by the number of Australian visitors who turn up complaining of the cold, you've gone from summer in the southern hemisphere to the northern tip of an island in the North Atlantic, in winter. What were you expecting? Tropical palms?)
I would say, don't bank on seeing much daylight. Some days it doesn't really get fully light at all. It doesn't bother me.
I'd recommend staying in and around the Shore area in Leith, great bars, restaurants and architecture, whilst avoiding the tourist traps and blatant rip offs in the city centre. Plus the 35 bus gets you into the centre in 15 to twenty minutes or so.
Also, I wouldn't bother paying to see the inside of the castle, it is way overpriced for what it is. It is worth going up to the Esplanade to get the view, but the castle interior isn't worth it. The Palace of Holyroodhouse is a bit gloomy in winter so not really worth the bother. Sumner time when the grounds are open, it is probably more worthwhile.
Going to see the Royal Yacht Britannia in Leith Harbour is much more worthwhile and better value (and I say that as an anti monarchist). The parliament is worth a visit too and the tours are great there.
But there isn't a bad time of year to come to Edinburgh, but my favourite time of year is April May or early June. But you can have a great time whenever you come.