Solution
aws s3 mb s3://[new-bucket]
aws s3 sync s3://[old-bucket] s3://[new-bucket]
aws s3 rb --force s3://[old-bucket]
Explanation
There's no rename bucket functionality for S3 because there are technically no folders in S3 so we have to handle every file within the bucket.
The code above will 1. create a new bucket, 2. copy files over and 3. delete the old bucket. That's it.
If you have lots of files in your bucket and you're worried about the costs, then read on. Behind the scenes what happens is that all the files within the bucket are first copied and then deleted. It should cost an insignificant amount if you have a few thousand files. Otherwise check this answer to see how this would impact you.
Example
In the following example we create and populate the old bucket and then sync the files to the new one. Check the output of the commands to see what AWS does.
> # bucket suffix so we keep it unique
> suffix="ieXiy2" # used `pwgen -1 -6` to get this
>
> # populate old bucket
> echo "asdf" > asdf.txt
> echo "yxcv" > yxcv.txt
> aws s3 mb s3://old-bucket-$suffix
make_bucket: old-bucket-ieXiy2
> aws s3 cp asdf.txt s3://old-bucket-$suffix/asdf.txt
upload: ./asdf.txt to s3://old-bucket-ieXiy2/asdf.txt
> aws s3 cp yxcv.txt s3://old-bucket-$suffix/yxcv.txt
upload: ./yxcv.txt to s3://old-bucket-ieXiy2/yxcv.txt
>
> # "rename" to new bucket
> aws s3 mb s3://new-bucket-$suffix
make_bucket: new-bucket-ieXiy2
> aws s3 sync s3://old-bucket-$suffix s3://new-bucket-$suffix
copy: s3://old-bucket-ieXiy2/yxcv.txt to s3://new-bucket-ieXiy2/yxcv.txt
copy: s3://old-bucket-ieXiy2/asdf.txt to s3://new-bucket-ieXiy2/asdf.txt
> aws s3 rb --force s3://old-bucket-$suffix
delete: s3://old-bucket-ieXiy2/asdf.txt
delete: s3://old-bucket-ieXiy2/yxcv.txt
remove_bucket: old-bucket-ieXiy2