You can actually do this with this rename command:
rename 'chomp(my $new=`sha256sum "$_" | cut -d" " -f1`); s/.*/$new.mp4/' *mp4
To preview what it will do before actually renaming, use -n. For example:
$ ls
'my movie file01.mp4' 'my movie file05.mp4' 'my movie file09.mp4'
'my movie file02.mp4' 'my movie file06.mp4' 'my movie file10.mp4'
'my movie file03.mp4' 'my movie file07.mp4'
'my movie file04.mp4' 'my movie file08.mp4'
$ rename -n 'chomp(my $new=`sha256sum "$_" | cut -d" " -f1`); s/.*/$new.mp4/' *mp4
my movie file01.mp4 -> 6a260f4adeb47000afeb9c53029ce0e14b6ea80be5c649ecdb155c8e7cfa1496.mp4
my movie file02.mp4 -> 14f24228ead10d639ef3db11ea09c6216aa80879327a2a57fb549e6474fb05b0.mp4
my movie file03.mp4 -> 6c9ecec6a75e217fa1786dd6f920a5c1db8473f17447595e0d33d7fc3530453a.mp4
my movie file04.mp4 -> 96a29ae5bc3a5e35537bcea1068c746ddba205d17fcc8e522a52402c7ed69927.mp4
my movie file05.mp4 -> 189207823129b57e52c86a743ea7154f65b8eccb5182f322abc3aae35f14057d.mp4
my movie file06.mp4 -> 93be0fd15be56bb22af94a0a0aa272d56f61e77160f1170305d41166d6135913.mp4
my movie file07.mp4 -> 05c92b5632414f14445af3facdb0e187746b3b0f40bad94a5f351aeb7f9a8847.mp4
my movie file08.mp4 -> 0d3cf9c165d4c18af8922b8daa9a6b1031c6762d179fb910b968d60af9aed255.mp4
my movie file09.mp4 -> da04afebaec6ea3d43dc3a0e66b790ab13751b00fd81aa1f882986c097c36298.mp4
my movie file10.mp4 -> 47bf532970d3e759346462a85d15ab9d9171c5a7688a6e90a1e976e071ecd167.mp4
And, to keep the original name:
$ rename -n 'chomp(my $new=`sha256sum "$_" | cut -d" " -f1`); s/.*/$_ $new.mp4/' *mp4
my movie file01.mp4 -> my movie file01.mp4 6a260f4adeb47000afeb9c53029ce0e14b6ea80be5c649ecdb155c8e7cfa1496.mp4
my movie file02.mp4 -> my movie file02.mp4 14f24228ead10d639ef3db11ea09c6216aa80879327a2a57fb549e6474fb05b0.mp4
my movie file03.mp4 -> my movie file03.mp4 6c9ecec6a75e217fa1786dd6f920a5c1db8473f17447595e0d33d7fc3530453a.mp4
my movie file04.mp4 -> my movie file04.mp4 96a29ae5bc3a5e35537bcea1068c746ddba205d17fcc8e522a52402c7ed69927.mp4
my movie file05.mp4 -> my movie file05.mp4 189207823129b57e52c86a743ea7154f65b8eccb5182f322abc3aae35f14057d.mp4
my movie file06.mp4 -> my movie file06.mp4 93be0fd15be56bb22af94a0a0aa272d56f61e77160f1170305d41166d6135913.mp4
my movie file07.mp4 -> my movie file07.mp4 05c92b5632414f14445af3facdb0e187746b3b0f40bad94a5f351aeb7f9a8847.mp4
my movie file08.mp4 -> my movie file08.mp4 0d3cf9c165d4c18af8922b8daa9a6b1031c6762d179fb910b968d60af9aed255.mp4
my movie file09.mp4 -> my movie file09.mp4 da04afebaec6ea3d43dc3a0e66b790ab13751b00fd81aa1f882986c097c36298.mp4
my movie file10.mp4 -> my movie file10.mp4 47bf532970d3e759346462a85d15ab9d9171c5a7688a6e90a1e976e071ecd167.mp4
The explanation here is a bit complicated as it requires understanding some basic Perl. The main idea is:
my $new=`sha256sum "$_" | cut -d" " -f1` : this will run the command sha256sum on each input file ($_ is Perl magic for "current thing"), and cut to keep only the hash. To illustrate:
$ sha256sum my\ movie\ file01.mp4
6a260f4adeb47000afeb9c53029ce0e14b6ea80be5c649ecdb155c8e7cfa1496 my movie file01.mp4
$ sha256sum my\ movie\ file01.mp4 | cut -d" " -f1
6a260f4adeb47000afeb9c53029ce0e14b6ea80be5c649ecdb155c8e7cfa1496
The chomp() just removes the trailing newline from the command's output. The result is stored in the variable $new.
s/.*/$new.mp4/' : substitute everything (.*) with the $new name.
Or, for the other command:
s/.*/$_ $new.mp4/' : substitute everything with the original name ($_) followed by the new one.