My sister in law is Bengali, albeit from Bangladesh, not India. There are Bengalis in India though. Be aware that India isn't a single nation/people/language, it's an artificial border that was drawn by British colonialists, there are actually many different languages and peoples in India. So rather than thinking "she's Indian" you need to think of what ethnicity she is within India. Also, consider whether her family are Muslim, Hindu or Sikh, or one of the less common Indian religions. That's where you need to start, and then work out the other details from there.
With names, someone's religion is as important, if not more important, than their first language (e.g. whether Bengali, Gujarati, etc). Muslims have Islamic names, Sikhs have Sikh names and Hinduism isn't just one religion but a whole bunch of religions that have existed in the region for millenia, predating Islam and Sikhism, all shoved together under one term. And there are Bengali names, Gujarati names, etc. Names often have honorifics. My sis in law's last name isn't a surname in the western sense, it's a name given to women, more like a title than a name. The Sikh names Kaur (for a woman) and Singh (for a man) are honorifics that baptized Sikhs take. They're more like Mr and Mrs than western surnames.
Bear in mind that the whole first name, middle name, surname thing is a western thing. Naming systems in other cultures can be very different. I'm most familiar with Islamic names (mostly how they're done in Arab cultures though). My sis in law is from a Muslim family and I've lived in two different Muslim countries for quite a few years and my ex partner is a Muslim. If your character is Muslim I can answer questions about Islam. I'm more familiar with Arab culture than Desi culture (i.e. India, Bangladesh, Pakistan, etc). Muslim names are quite straight forward when you get the hang of them.
What you want to avoid is combining the wrong names. Like if you combine Sikh, Hindu and Islamic names, e.g. Abdul Ganesh Singh. The first being Muslim (actually it's only half a Muslim name because Abdul is never a name on its own, it's always followed by one of Allah's 99 names), Ganesh is a Hindu name and Singh is a Sikh name. Also, don't assume that all the family will have the same last name like in the west. Muslim women keep their family name upon marriage, and some last names are honorifics rather than western-style surnames.
Also middle names is a western thing. In Arabic culture, your middle name is your father's name, so women and girls have a male name as their middle name, e.g. a Muslim woman might be called Aisha Muhammad as her first two names. In this case, her dad's first name is Muhammad. She'll keep her dad's family name after marriage and not change it to her husband's. I'm not sure to what extent this is done with Desi Muslims, but I can find out.
