How can a Muslim girl convince her family – who are of moderate religious observance – to allow her to wear a loose abaya, head covering, gloves, and niqab, despite their objections and excuses of her being too young, or fearing people's opinions, or claiming the niqab is not obligatory?

To be good to your relatives, you should pray for their guidance, look upon them with compassion, treat them with good character, and maintain ties of kinship through financial support and other means. Avoid arguing with them initially, and try to convey advice to them indirectly.

The hijab is an obligatory duty for the Muslim woman, and it is not permissible to abandon it for the sake of family. Covering the face is obligatory according to the most preponderant opinion of the jurists, and there is no obedience to anyone in preventing it. However, if wearing the niqab leads to you being prevented from wearing the basic hijab, then there is no sin in temporarily abandoning it.