Well, the producers have their reasons for axing the public vote, but I think they could introduce the PUBLIC VETO. As usual the HoH nominates two people and then the public vote to save one of the nominees. All housemates except the HoH then compete in a Veto Challenge, with the winner choosing the replacement nominee. The housemates then vote between the remaining nominee and the veto replacement.
When three housemates remain it would then be the public that decided the winner. Alternatively, a final competition would determine the last HoH. The evicted housemates would then choose which of the other two housemates go into the final, with the public deciding the winner.
Other alternatives are like said in the opening thread - a BB3 style system as used in week 1. All housemates except the HoH face the public vote for eviction, and the housemates decide which of the two with the most votes are evicted. Therefore, the producers get their moment of truth in the live show.
And obviously, the easiest way to bring it back is for the HoH to nominate and the public to decide who is evicted. This is the system I'd use if it was introduced into the UK - I think introducing the Veto would initially be too complicated as producers would want to keep the nominees secret until the day the lines open.
However, the HoH could nominate on a Monday. The Veto competition could take place in the afternoon with the winner being secretly informed who is nominated and whether to use the veto in the diary room. The HoH then makes a third nomination in the diary room (anyone but the Veto holder) but would not know if the veto had been enforced. The final two nominees are then revealed on Tuesday.