Gargoyle,

Wyrd Sister and you have it right: the personality mix of teacher (patient, nurturing, appealing and enthusiastic) and student are the essential firsts.

Then you might want to think about the sort of music you want your daughter to play. My own d. started Suzuki cello at around age 8 with a wonderful teacher (actually, teachers since we moved to another city in her second year of lessons). Suzuki is oriented toward classical, but it is a great foundation for music/performance in general, and after a year of Suzuki my daughter picked up fiddle on her own because Old Timey was the music played most often around our house and because I got her a 1/2 size violin which I made available but didn't force into the Suzuki/cello practice time.

Web page for the Suzuki association doesn't list violin teachers in Phoenix (http://www.suzukiassociation.org), but the list is not at all complete.