I understand. They are claiming that you used an improper card to enter the subway. This actually does qualify as Theft of Services. However, they have to prove an essential element of the offense. They have to show that you attempted to deceive your way into the subway. The important thing here is that the State has the burden of proof to show that you knew of the deception. Unless you gave statements to police to the effect of "Yes, I knew it was my mother's, I just wanted to save money" then it would be difficult for the State to bear the burden here. If this was an accident, the law will not hold you criminally responsible for this offense at least.
However, regardless of the first issue, you may qualify for an expungment or sealing. With minor, non-violent "offenses" like these, and a clean criminal background such as you have described, an attorney may be able to negotiate any one of many outcomes wherein you can qualify to remove the arrest and ticket from your background.
Because expungment and sealing laws are so state intensive (i.e., every state can be drastically different when it comes to qualifying for an expungment or sealing) it would be very prudent to discuss this portion of your questions with a local attorney who is very familiar with the requirements in place in New York. This is so because, although some outcomes in the criminal case may seem favorable initially (for instance, no fine), the actual outcome may affect your ability down the road to remove the arrest and charge from your history.
I hope that I have been able to assist you. If you have any follow-up questions, feel free to ask. Otherwise, please leave positive feedback. Thank you!