The final scene on the bridge was originally somewhat different. It was all filmed, but the first section of it was edited out and then changed with the addition of Kirk's new captain's log voiceover.
I think they made the change after a preview screening, feeling the tone needed to be more reflective.
Here's the scene as originally written:
INT. ENTERPRISE BRIDGE 247
The door whooshes. Kirk strides on the bridge, David with him. Carol is there with Bones; they are glad at what they see.
SAAVIK
Admiral on the bridge!
KIRK
As you were, Mr. Saavik.
SAAVIK
Aye, sir. On course to Ceti Alpha.
All is well.
KIRK
Good, I believe you already know
my, uh, son --
Saavik comes close to blushing --
KIRK
(continuing)
Yes, well, why don't you show him
around and...
SAAVIK
Aye, sir --
She takes him off, deadpan amusement --
SAAVIK
(continuing)
So you turn out to be the dumb
bastard.
They stare at her.
SAAVIK
(continuing)
That is a little joke.
Kirk has a smile. He joins Bones and Carol facing the screen.
BONES
She's changed her hair again.
(pause)
Will you look at that? Think
they'll name it for you, Doctor?
CAROL
Not if I can help it. We'll name
it for our friend.
KIRK
(quietly)
It is a far far better thing I do
than I have ever done before... a
far better resting place I go to
than I have ever known...
CAROL
(soft)
Is that a poem?
KIRK
Something Spock was trying to
tell me. On my birthday.
He shakes his head, remembering --
BONES
You okay, Jim? How do you feel?
KIRK
(considers; smiles)
Young. I feel young, Doctor.
He turns to Saavik and David --
KIRK
(continuing)
Lieutenant, new course: 'Second
star to the right and straight on
till morning.'
The bridge crackles with lights and CHATTER as we...
CUT TO:
247A THE VIEWSCREEN 247A
AND GO THROUGH TO:
248 EXT. SPACE FINALE (BOARDED)
SPOCK'S VOICE
Space, the final frontier... These
are the continuing voyages of the
Starship Enterprise... Her ongoing
mission: to explore strange new
worlds... to seek out new life and
new civilizations... To boldly go
where no man has gone before...
She is moving out now, passing CAMERA and heading
toward the distant stars. She is beautiful and they are beautiful. And as she slowly disappears from VIEW...
MUSIC RISES. CREDITS ROLL.
THE END
Meyer also talks about reshooting the scenes in the corridor leading to the Genesis cave. The shot a different version but came back after filming was finished to redo it.