Anjali comes from the Sanskrit word meaning "gift" or "offering". Like all Indian names, the accent is on the first syllable: ANjali, like MADHavi or even like JERemy (see, we have a family thing going here).
So now we get to the pronunciation of the "An" part of Anjali. The "An" is pronounced "UN" as in Unjali. This "An" one would find in the name "Anita", This is in contrast to the "An" being pronounced as "AWN" like in the name "Anika". The problem of the pronunciation occurs when one of takes Indian sounds and uses Roman letters to represent them. And aesthetics play a part too. Some people spell Anjali with a U so it's 'Unjali' but we weren't too keen on that. Some people spell Anjali with a U again, as in 'Anjuli' but we weren't keen on that either. And we definitely weren't keen on 'Unjuli' either. (Two U's - oh no!) So dear friends, think UN-JALI for Anjali.