Sun should cooperate, heat - well maybe not hot but it should be pleasant. Normal temps : Mid to upper 60's during day, 40's at night although they are predicting La Nina for this winter, which will mean the temps could be higher than normal. The temps will be almost the same in Tucson [and I assume that is what you mean by SOUTH]

http://www.fodors.com/world/north-america/usa/arizona/phoenix-scottsdale-and-tempe/

I have been several times in Tucson in late December, and each time, the weather was just lovely. There are beautiful resort hotels in the foothills north of central Tucson; we particularly liked Westward Look Resort. Once, we spent the wee hours of the New Year in a hot tub under the stars of the Arizona sky...

There is much to see and to do around Tucson, especially with a kid:
- The Arizona-Sonora Desert Museum (which is more like a zoo)
- Saguaro National Monument
- Tombstone
- Mission San Xavier del Bac
- if you like: Biosphere 2
...

My trip report from earlier this Spring - http://www.fodors.com/community/united-states/sa-kids-are-back-from-spring-break-in-az.cfm
There is another post about the same time asking similar questions which may also be helpful

Weather will be quite nice - relax.

Pack jackets and jeans as December and January can be quite cool. Tucson is cooler than Phoenix as it's at a higher altitude.