Where in Italy to do a good, cheap walking/hiking trip?

I only know of one hostel in that area and it appears to close during the winter.
http://www.brikette.com/index.html

This campground near Pompeii has bungalows really cheap, but you would have to check and see if they are open at that time of year.
http://www.campingspartacus.it/

You might want to get some type of transportation pass for the area.
http://www.unicocampania.it/index.php?lang=en&center=inside&colonna=turisti&action=costiera&banner=costiera

The bus (Marozzi or Curreri) between Rome and Sorrento will be cheaper than the trains and take about the same time. They only run about once a day in winter. Cost is about 16€.

http://www.discovercilento.com/

You should also consider le Cinque Terre, even though it is a further train trip from Rome. Because of geography, it actually gets an abundance of sunny weather and some fairly mild temps in winter -- although no guarantees. But it is a popular hiking destination year round. Even though it is touristy, it is not as touristy as the Amalfi and it has a lot more budget accommodations in the five towns, which get even cheaper if you stay outside the five towns, or pick the interior hills instead of the sea. You should be able to cut a deal for a long stay in any b&b. Be sure to get a rock solid you will have heat wherever you go, north or south. Here are some possibilities:

http://www.dreamland-bb.com/

http://www.chincamea.it/DefaultEN.aspx

http://www.rifugiouscio.com/

These two are closer to Genoa:

http://www.larosabiancadiportofino.com/

http://www.rifugiouscio.com/

If you are interested in Cilento (which would be my first inclination), you might read up on these kinds of itineraries to help you plan:

http://www.sherpa-walking-holidays.co.uk/tours/inntoinn/jcndos.htm

http://www.nonsolocasa-italy-travel.com/Holiday-homes-Cilento/Cilento-holiday-homes.htm

http://www.morigerati.net/servizi_extra_eng.asp

If you want to stay directly in the Amalfi, check out the town of Atrani, and also the hills away from the sea in Sorrento -- but again, make sure you get a guarantee of heat. Most of the area really shuts up tight in the winter.

http://www.guardian.co.uk/travel/2009/aug/09/cilento-italy-amalfi-castellabate-acciaroli

I just found this webiste and you might find it very useful. You can use it to search out accommodations on wineries. For the Amalfi and Cilento areas, search for Campagna. For le Cinque Terre hiking, search in Liguria. Some of the accommodations are quite cheap, and the descriptions note if there is heating. Probably some of the wineries would cook you meals, but if not, and you don't have a car, you need to ask how far it is to the nearest restaurant and make sure it will be open during your stay.

http://www.vinogusto.com/en/winetourism

I spent ten or so days in Italy, in January, two years ago. It was COLD. Much colder than my recent London trip in February. We were in Orvieto/Assisi/bits of Tuscany. I was well dressed (including nylons under my trousers, and silk camisoles under the sweaters and a thick wool coat and scarf/gloves) and was still freezing.

Oddly enough by the time we made it back around to Florence it became tolerable but it was still not what I'd call hiking weather.

Tuscany and Umbria are indeed colder than places on the Mediterranean, but again, no guarantees.

You should really give Le Marche a go, similar to Tuscany and Umbria but without the tourists and the high costs. The Sibillini national park has amazing hikes, try this link for more information http://www.walkingworld.com/home/index.asp?id=45 and try a stay at funky www.villasanraffaello.com as they are into serious hiking, you can walk into the national park from the villa and they supply you with free itineraries

In late December/early January, Le Marche will be extremely cold -- and I mean extremely cold -- and the hikes of the Sibillini impassable and closed.