I would bet the WH told her to do this. Obama may have been setting her up, of course, but I think it was really because he had no intention of letting her be a real Secretary of State (since one of the reasons she was always off partying in Brazil or wherever was that Obama never let her go on any serious visits, and only summoned her into the public eye when he didn’t want to take the heat for something).
He didn’t want any official connection with her and told her to set up a private account. There’s no way a Secretary of State would have done this otherwise.
She would certainly have been fine with it, of course, because the Clintons have always been weird and secretive, but also because she seems to have been genuinely intimidated by Bambi.