I have a simple bucket that looks like images.mysite.com on my S3 and other buckets containing backups, etc.
I want to allow a specific user to be able
I am interpreting this question as: "Can I allow access to one bucket where any other buckets will not be accessible and thus invisible." Because, showing the name of the bucket to which no access was granted still equates to information leakage.
And the correct answer is no. The required permission is ListAllMyBuckets which will allow the user to see ALL buckets. Leaving out this permission will make the console unusable.