As Alex said, it is one of the biggest weaknesses in SharePoint.
You need to decide when and how you are going to reduce the size of the list. And put a governance policy (written document that is vetted through management) in place that states that lists need to be below the default 5000 item limit.
I DO NOT recommend increasing the limit, although I have seen companies do this. All it leads to is the eventual crash of your farm and many, many more lists that then need to be paired down.
to fix your issue though, you could increase the limit, then export the list to excel, then decrease the limit back to default 5000 and then do your work in excel by parsing the list into however many you want, based on some criteria that you decide (date,
unit numbers, or what ever makes sense)
then import the excel files to new lists, after you have the new lists imported, business could test, then you could increase the limit one more time, so you can delete the original list.
and for renaming the site, if you don't want to do all the list work, you could increase the limit, and rename the site with set-spsite, assuming you're using host named site collections. If you're using path based, you could export the site, then
create the site and all its subsites, and perform the import; which is one more reason why HNSC is the way