The unknown in this question is, "Where will Harry be when Ebenezer dies?" If Harry just get's word of Ebenezer's death or if Harry is present, but so are a number of powerful members of the White Council, then in all likelihood the Council decides and either Captain Luccio or Warden Ramirez gets the Blackstaff. The only reason I don't think it will be Luccio is because her fighting ability has been diminished. The Blackstaff would make her uber-powerful, but it's a weapon that should only be used with great discretion and only at times when nothing else is appropriate. So, I could see Ramirez getting the job in an attempt by the Senior Council to counter-balance Harry.
On the other hand, if Harry and Ebenezer are together when Eb meets his end, but no one else is around, that changes things. I could see Harry picking up the staff and wouldn't you know it, Mab shows up to give Harry the most dangerous mission he has ever faced. A task that even the Eye of Ethniu would be of little use in completing. Now Harry won't want the Blackstaff, so I could see Harry getting summoned by Mother Winter and getting told that either he has to use the Blackstaff to complete his mission or return it to it's original owner, Mother Winter herself. It doesn't have to be exactly that scenario, but it could be something close to that.
If Harry does pick up the Blackstaff, my prediction is that it will be a one and done affair. As Harry is no longer a member of the White Council, he couldn't act as Blackstaff Dresden of the Council, though it would give the Council a new level of deniability if Harry did their dirty work for them. However, most of the Senior Council would never trust Harry with that job. After using the Blackstaff once, my guess is that Harry would return it to the Council rather than Mother Winter. Then again, maybe not, I don't know. It's just a feeling I have.