There's no way to do this for Free and entry-level ($10) Cloud systems easily - you have to register for the new domain you want (which will of course, still be <something>.atlassian.net), export from your current system and restore the backup into the new one.
For the higher tiers of system, you can raise a support request with Atlassian, with your current and desired names and they'll change it from <current>.atlassian.net to <desired>.atlassian.net (but they won't do it frequently, and obviously not if <desired> is already taken)
For JSM June Challenge #2, share how your non-technical teams like HR, legal, marketing, finance, and beyond started using Jira Service Management! Tell us: Did they ask to start using it or...
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