Currently I have a structure as below:
I need to delete and re-create the site "thesite" but I don't want to lose the site "my".
Is this somehow possible?
If there are many such cases, then I would suggest you to use Metalogic Migration tools, if this is one of the case then , following should help.
Take a full backup of the subsite (http://thesite/my) (MySite)
Delete the sitecollection in central admin, take a backup of whole site collection before u delete , to be on safer side :)
recreate the root sitecollection (http://thesite) the way you want and restore the subsite back
Hope it helps
You could save the "my" site as a Site Template (including content) and then importing the resulting .stp/.wsp into the Site Template Gallery of the new "thesite". Then, create a site based on that template.
That would work if the "my" site is not heavy on data.
Another option if you do not have much content in your site is to use:
Site Actions > Site Settings > Content and Structure (http://thesite/_layouts/sitemanager.aspx)
Right-click the site and select Move