Hi,
The simplest and safest way is to delete the data from both DSO and InfoCube. Then you add the new Characteristic through the complete data flow (i.e. Transfer Rules, Update Rules, InfoSource, DSO, etc). Afterwards you reload the data.
For reloading data, I suggest to do it in yearly slices if you have an appropriate Time Characteristic. Suppose you have Fiscal Year/Period. You can do a delta initialization starting on Fiscal Year/Period 001.2016. The older Fiscal Years you can upload retrospectively via Repair Full Requests (w/o impact on the delta process).
Best regards,
Sander
PS: Maybe also a good timing to upgrade the data flow to 7.0 data staging technology?