Question asked during the August 30th, 2012 webinar titled "Improving Processes with Scrum and Kanban"
The rapid board does not provide an auto means of doing this.
A suggested process would be to clone the story in JIRA and edit the clone for the remaining work. If the Product Owner prioritised this work was to be continued, this would then be sized and estimated in sprint planning.
If you clone the stoiry there is an otpion to copy the sub tatsk as well. This can be useful if there are a lot of unclosed sub tasks but it means you have to delete the suib tasks on the original story. The other option is not to clone thje sub tasks but move / reparent them to the new story.
The story would then be moved from your backlog to the next sprint board as normal.