There are so many places to visit around Rota I really wonder why you ask...The best base to visit Sanlúcar de Barrameda and Doñana if you like nature; Jerez for horses and wine cellars; Puerto de Santa María; Cádiz... baroque and good seafood everywhere. Possible daytrips to Arcos-Bornos or MedinaSidonia.

You will love Spain!!! You are going in the off season wich can be a little disappointing at times especially in a small town. I went to Zaragoza, Tarragona (a small town outside of Barcelona)and Mallorca over Christmas break a few years ago and I did see alot but there wasn't much atmosphere in the streets and many restaurants and shops were closed. The smaller town was a little like a ghost town. Since you are visiting family your small town time should be enjoyable and they will know were to go at this time of year.
After visiing you may want to explore the rest of Andalucia, it is spectacular. Sevilla (el alcazar, a real flamenco show maybe a bull fight), Granada (La alhambra), Cordoba and Rhonda (just a beautiful town) would be my top suggestions. In January it won't be sweltering hot which should make for nice exploring weather. You could see many beautiful historical sites and it would be less expensive stay close by.
I found a site with a sample itenerary of these places check it out and good luck I think you will have a great time.
http://www.totallyspain.com/spain_travel_offer.asp?id=57&gclid=CMjo15v2kqkCFchg2godnz1Dpw

Dig deep into the area around Rota, as josele suggests. Here's a way into non-touristy, good-humoured and liberal Cádiz (three thousand year old Phonecian Gadir) state of mind: http://www.youtube.com/watch?v=33JUrGwZQLY

You will find yourself in the heartland of flamenco culture, and much more so at this time of year. Not far from Cádiz is Sanlúcar de Barrameda. Have the freshest seafood and an ice cold manzanilla in one of the seafood restaurants along the river bank of the Guadalquivir. They only make this fantastic wine here in Sanlúcar, like a very dry sherry (Jerez).

The greatest flamenco singer of all times, Camarón de la Isla, was from Cádiz (San Fernando). This song revolutionized flamenco song when it was released in 1979: http://www.youtube.com/watch?v=l5N3T8w4Zvc

The most famous flamenco dancer in the world today, Sara Baras, is also a native of Cádiz: http://www.youtube.com/watch?v=OQ-ie12Eqck

And Paco de Lucía, the greatest flamenco guitarist ever, is from your region (Algeciras): http://www.youtube.com/watch?v=2oyhlad64-s

Thank you very much for your ideas! I have looked into all your ideas and everything looks close enough for short day trips.

Spend a week in Seville and visit Granada for a night or two, you won't regret it.

Dear Jessie,

Today I joined for the first time the forum. As you are a little bit confused about Spain, especially Andalusia, I must tell that you really have to come and visit Seville. I`m Dutch but living 15 years in Seville. I travelled a lot around Europe but Seville and Granada is lovely.

Best regards,
Natasja