You can use "clone" from the menu options on an issue.
But for a regular repeated task, it's probably better to write a very short script that uses REST to create a new issue (Pi + Cron + 20 line script, 19 lines of which is poking me on HipChat if the create fails)
There is the functionality built in it is called 'Clone' and it can be found under the 'More' actions on any issue provided the user has create permissions.
However, as you say you have a set of tasks to be repeated every week you may wish to look at one of the scripting plugins to make this easier for you as you could then clone multiple tickets at once and bulk populate with certain parameters.
Hope this helps.
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