What was Jon Snow's mother's name?

You're asking a colossal question. This link will guide you through all the possible answers to that question. Do note, there are major spoilers afoot there.

To summarise some of the possible information:

In the books, we are presented initially with the idea that Ashara Dayne is his mother. According to Ser Barristan Selmy, a Stark bedded her at a jousting tournament. She was pregnant, but eventually killed herself by jumping from a cliff. According to Ser Barristan the child was still-born (giving a reason why she may have jumped), but one theory is that the child was actually Jon Snow. Cersei, for example, believes this.

Another theory is that Wylla was the mother. She was the wet nurse at Starfall. Robert recalls Ned bedding a "wench" during the rebellion and questions Ned about this - he answers it was Wylla but says no more. Some, for example Sir Edric Dayne, believe this to be true - Sir Edric later tells Arya that Wylla is Snow's mother.

However, neither of these are the prevailing theories amongst most GoT fans. As the presumed truth is extremely spoilerific, I'll enclose it in spoiler tags:

The prevailing theory is that Jon Snow's mother is Lyanna Stark, from a union between her and Rhaegar Targaryen. There are a few extremely strong hints for this in the books. For example, Ned Stark continually remembers Lyanna's "promise" to bring her body back to Winterfell to be with her family. But as the first novel goes on, he keeps remembering this promise when thinking of Jon Snow - indicating perhaps the promise was much more significant than that. Additionally, when it is considered the King's guard were protecting Lyanna, it seems unlikely she was a captive of Rhaegar's, and much more likely she was his partner. Ultimately, this would make him a Targaryen and would tie in with Daenerys' visions in the House of the Undying, where Rhaegar is shown with a son, Aegon, who appears later in the novel series. However, in the House Daenerys is told "there must be a third" - as the dragon must have three heads. This strongly suggests it could be Jon Snow. The only person who knows for sure is Howland Reed, as only he and Ned Stark were in the room with Lyanna when she died and passed this information on. With Ned dead, Howland is the only person who still knows. Note: You may or may not have noticed on fan sites the "formula": R + L = J. That is where this comes from. It'a a spoiler-free way of saying Rhaegar + Lyanna = Jon.

In conclusion, this question has not been definitively answered and will not be until future books are released (or the TV series overtakes the books). However, the answer in the spoiler is the overwhelmingly supported theory.

Edit

As this has been discussed in far more detail elsewhere, I'll also direct you to this amazing online forum for A Song of Fire and Ice. That link will take you directly to spoilerific content, but has plenty of answers and an in-depth analysis far beyond what I've provided above to explain the origin of Jon Snow.

In season 6. episode 9 "The Winds of Winter", it is revealed that mother of Jon snow is Lyanna Stark.

From Wikipedia

Bran Stark (Isaac Hempstead-Wright), Meera Reed (Ellie Kendrick), and Benjen Stark (Joseph Mawle) arrive near the Wall. Benjen takes his leave, stating that the Wall's magical protection prevents him from being able to pass. Meera assists Bran to a weirwood tree and he re-enters the vision of Eddard Stark (Robert Aramayo) at the Tower of Joy. Ned finds his sister Lyanna Stark (Aisling Franciosi) covered in blood from childbirth. With her dying breath, Lyanna pleads with Ned to take care of her son, who is revealed to be Jon Snow.