Tolkien himself never mentions Legolas's mother within the books, only that of his father, Thranduil and his grandfather, Oropher (who are Sindarian Elves).
However, The Hobbit trilogy expanded more on Legolas Greenleaf's mother, however, this is only within the movies, not the books (so it technically can be considered Peter Jackson's own work of fan fiction). But in Desolation of Smaug and The Battle of the Five Armies, we are given some mentions of Mrs. Thranduil.
The White Gems of Lasgalen, though only mentioned in the behind the scenes featurettes, was originally Mrs. Thranduil's jewelry, thus explaining Thranduil's motivations in retrieving the "heirlooms of his people." As apart of the backstory written for the film trilogy, the jewels and necklace were fashioned for her but she never got to wear them.
There was also a cut scene from BOTFA that included Gandalf telling Thranduil:
Gandalf: Those gems weren't all your wife left you, my friend. She left you a son. Tell me, which would she have you value more?
In another scene kept in BOTFA, Legolas explains to Tauriel that his mother was captured and brought to Gundabad where she died after enduring torture, which many people have misinterpreted to mean that she died in battle.
Legolas: My mother died there. My father does not speak of it. There is no grave, no memory. Nothing.
We do have an image of her as a statue in The Desolation of Smaug, which was confirmed to be Thranduil's wife in the behind the scenes featurette of Thranduil: The King of Wood and Stone:
It was also implied in this BOTFA scene that Legolas's mother had been captured while defending him in battle:
Thranduil: Legolas, your mother loved you...more than anyone...more than life.
Though we are offered this additional information, we still do not know if she was Sindarian or Silvan, her name, or how she met Thranduil. It is also fair to note that this was what Peter Jackson and his team came up with for Thranduil's backstory, not what Tolkien himself wrote. Tolkien never mentions Thranduil's wife ever in his written works.
So it is either left entirely to the imagination or we can go by The Hobbit
trilogy (though some would accuse it of being a little too flexible and carefree with Tolkien's timeline). If you wish to look at Thranduil's backstory from The Hobbit trilogy, there is a behind-the-scenes featurette that you may watch for more additional information on Thranduil's wife and the mother of Legolas Greenleaf that further discusses this topic.