The people closest to Leia (in-universe) and that appear later (out-of-universe) all pronounce her name the same way: LAY-uh.
However, it does seem at the time A New Hope was filmed and released the pronunciation may have been; LEE-ah.
In the scene where Padme gives birth, Obi-Wan asks her to name the children. At about 0:47 in the video Leia is born. Padme clearly pronounces it LAY-uh.
Episode IV Just prior the destruction of Alderaan, Leia and Tarkin have an exchange. Tarkin pronounces the name LEE-uh
Destruction of Alderaan on Disney Video
Also, Senator Dodonna uses the LEE-uh pronounciation during the briefing prior the assault on the Death Star. Heard at about 0:11.
Luke calls out to Leia vocally but is trying to use the Force. He uses the LAY-uh pronunciation. See this around 0:50.
When Luke is talking to her about their mother he pronounces it LAY-uh. This occurs at about 0:04.
When Leia and Han reunited, Han pronounces it LAY-uh at about 0:30.
Out-of-universe: Episode IV was the first film released and seems to be the only place where the name is pronounce "Lee-uh". Newer movies all have the same pronunciation, so it appears that a decision was made to either change or standardize her name after the first film.
In many interviews from the time period the pronunciation is also LAY-uh by the hosts and the actors: here is one example of Carrie Fisher doing just that at about 1:07