If you are looking for cheap new year is perhaps not the best time.
The flights go sky high and most apartment owners put up there price for the Xmas and New Year fortnight.
Mid December is very reasonable as is January when new year is over.
Mid December weather is normally better than Mid Jan.
Sorry but this is fact.
Rodz

Sorry Rodz, but I have found that Christmas is expensive. But dont forget they dont celebrate untill 6th night, 5th January. We flew out one year on January 1st. This is when the flights are a lot cheeper than a few days before. Like us your children will be able to visit one of the 3 Kings, they will need to take a small wrapped present, labled either boy or girl and the age. They will exchange this for a sweet. On the eve of 6th night there will be a nativity when the 3 Kings come to the baby Jesus. Followed by a firework display. At mid night local people will exchange presents with a night of celebration. When my own children were young we did this many times, a traditional Christmas at home followed by a Christmas in the sun. We always stay in Puerto de la Cruz.