Mentioned in In the Cage: A Guide to Sigil, page 39, Li'kore is a "githyanki princess" searching for her exiled mother who escaped death at the hands of the Lich-Queen. She seeks to present her queen with her mother's head in order to to earn command of an astral fortress.
That's all the information on Li'kore or her mother that exists, as far as I know.
In what sense do you think Li'kore is a princess?
1. She's not a princess. Her mother a knight, commander of a fortress, high enough level to merit the Lich-Queen devouring her soul. Li'kore is a another knight looking to prove her loyalty. She may have only recently discovered what happened to the woman who laid the egg she hatched from. The high-ups at the Fortune's Wheel think of her as a princess because they don't understand githyanki culture.
2. She is a princess. Her mother was Vlaakith's older sister a thousand years ago and Li'kore is, in fact, the last living inheritor of Vlaakith's line.
3. She's sort of a princess. Her mother was the false githyanki queen described in the Dark Sun adventure Black Spine.
If she's the last of the Vlaakiths, then the Githyanki suddenly have more of a future if/after the Lich Queen dies.
I like this line of thought, anyway. There is no such thing as the last of some royal family...there is always this or that bastard child with a claim to the throne, somewhere.
Even if the bloody hag has had a millenia to get rid of her rivals. After all, you don't age in the Astral plane.
Option 4: She's sort of a princess. Her mother is Zellor, the githyanki knight from Vortex of Madness.