in most churches, you have to wear a veil (as a woman) before taking Holy Communion, and it is recommended during liturgy. some churches have loose lacy veils, others have complete veils that cover head and neck (esp. ethiopian and eritrean churches) and most of us are somewhere in between.
if you are not orthodox, or if you are brought up orthodox but are now unsure, and are not communing, it is certainly not necessary to wear a veil. it's fine for visitors and enquirers not to veil.
i agree with katherine of dixie, presumably there is some culture clash or generation clash with your friend and her mum.
if elders insist on keeping all the small rules and traditions but are bad at the big things (love, forgiveness etc) then it can leave people hating the tradition. i used to think i hated all tradition.
then i found the orthodox church and found the right tradition brings real peace