I too have heard "Ah-nee". Show the doubter this thread, and unless he can get Ani herself to say otherwise, we've got him way outnumbered.

Although Ani and her family could choose from among various possible pronunciations of their names, Ah-nee DiFrAHnco would be in line with European (in this case presumably Italian) pronunciation, as opposed to Annie DiFRANKo (rhymes with bank), which would be an Aglicized variant. There may well be some DiFrancos and DeFrancos who use the "bank" sound, but in Italy (or Spain or France or...) the "a" is pronounced like the "a" in "father".