One full side of my family, also Jewish, was born in Romania, and I know its not easy to find the town or area of origin.
Finkelstein is a very common name, so it is important you try to narrow down the birthplace.
What worked best for me was first finding where the family settled in North America. Generally they would not have come alone, but would have come with other Jewish families. Often if the families came at the same time, they could have come from the same place. So find out more about the other Jewish families in the area. Maybe see if other genealogists are researching them and contact them to find their origins.
Second step: Get the ship immigration record. Find out when the family came to North America and which port. Identify them in the record and it will always give the location they came from - usually the province in Romania, which will narrow things down. Again, look for others on the same ship from the same area and find out about them.
Third step: Find census records of your early immigrant ancestors. Often the Census will give the place of birth. Were any married here, or died here? Marriage and death records rarely list place of birth as more than Romania, but you may get lucky.
Fourth step: Join the ROM-SIG of JewishGen, and look through all of their resources. There is a lot of information available here. Don't assume any of the Finkelsteins you find are related until you have enough evidence to show it.
No one said it would be easy.