The realm of Thranduil in Northern Mirkwood is known as the Woodland Realm, or more commonly, the realm of Thranduil.
In the North also there had been war and evil. The realm of Thranduil was invaded, and there was long battle under the trees and great ruin of fire; [...].
The Lord of the Rings: Return of the King - Appendix B: The Tale of Years
Thranduil, king in the north of Greenwood the Great, was one of these. [...] Thus in one essay Thranduil's realm is said to have extended into the woods surrounding the Lonely Mountain and growing along the shores of the Long Lake, [...].
Unfinished Tales: Chapter III - The History of Galadriel and Celeborn: Appendix B - The Sindarin Princes of the Silvan Elves
So it was that Sauron prepared two strokes- in which many saw the beginnings of the War of the Ring. They were made together. The Orcs assailed the realm of Thranduil, with orders to recapture Gollum; [...].
Unfinished Tales: Chapter IV - The Hunt for the Ring: Of the Journey of the Black Riders according to the account that Gandalf gave to Frodo
Its capital is called the 'Elvenking's Halls', also known as the 'Halls of Thranduil'. This realm did not take any name of Sindarin dialect, as the Silvan Elves did not speak Sindarin, even though some of their ancestors were of Sindarin origin.
Thranduil's Halls are named in the map: