For good weather and festivals, late January
India (all Tamil Nadu) Karnataka Pongal harvest festival, 3 days
(Madurai) Float festival, late Jan/early Feb.
(Jaisalmer, Rajasthan) Desert Festival. Touristy but terrific location, under the full moon. Dance, music and camel riding if you must. Nearest airport is at Jodhpur, 280kms away.
Vietnam (Nationwide) Tet (new year) Jan/Feb 1 month. Vietnam Pictures.
Malaysia (Kuala Lumpur) Thaipusam (masochism fest)
(Penang) Chor Soo Kong (snakes), 6 days after Chinese new year. Malaysia Pictures.
Philippines (Panay Island) Ati-Atihan, kind of Mardi Gras, 1 week
Guatemala (Esquipulas) El Cristo, 1 week around 15th.
Canada, Quebec City, Carnaval de Québec, 10 days of parades, themed parties, ice sculptures and winter sports of every description, in a pretty Francophile city. Late January- mid February.
Or go for festivals only, not the weather! i.e. this is not the best season
Turkey (Selcuk, Ephesus) Camel Wrestling, 3rd weekend of January every year. One day only. Turkey Pictures.
Bolivia (Copacabana) Candlemas. Late Jan-Feb
Peru (Puno, Cusco+) Candlemas. Late Jan/early Feb. 2 weeks.
(Cusco) Dia de los Reyes, Jan 6.
China (Harbin) Ice Lantern festival. All January and February. Very spectacular, very cold! China Pictures.
Ecuador (San Lorenzo) Santos Reyes, painted kids. Jan 6.
(Cuenca) New Year, up to Jan 6.
Nepal, Khatmandu Valley, Basant Pancha festival, 1 day.
Japan (Osaka) Toka Ebisu, Jan 9-11.