His eldest surviving uncle on the Austrian side and the Prince of Coburg on the Hungarian? A bishop from either side? Along with the real movers of policy, Beust, Andrassy, Deak.
They would defeat the federalist plans of Hohenwart and effect a reconciliation with Prussia. They would repeal the concordat, promote Dualism, quasi-democracy and economic growth.
Archduke Karl Ludwig would be certainly on the Regency council (and probably also his younger brother Ludwig Viktor, although his proclivities may be quite an obstacle). Andrassy and Beust (both Minister-President at the time) would be on the regency council. Deak is certainly a good suggestion. Albrecht of Habsburg-Teshken is another strong candidate (both as a conservative and a representative of the army, not to mention he's a relation); however he's also a fire-breather (proposed both a preventive war against Italy in 1876 and the "dash to Salonika" in 1878) who will have to be reined in. What about the Dowager Empress (Sissy)? Would she be on or not?
There must have been a decree governing the composition of the Regency council, but I was not able to find it.
There will be quite a long Regency (Rudolph is just 10 in 1869) and interestingly the first revision of the Ausgleich (1877) will have to be dealt with by the regents. Guess that the Hungarians will try and gain more concessions.
It will be a council divided along political (liberal vs. conservatives) and even more importantly ethnic (Magyar vs. German, and both vs. Slavs) lines and I don't believe it will be capable of promoting reforms: the only point on which everyone would substantially agree is the rapprochement with Prussia (and this mostly for lack of viable alternatives). Obviously a lot would depend on Karl Ludwig, who will certainly preside the council, but from what I know Karl did not have a strong character and was not much interested in politics.
Not much hope for liberalization or democratization, and even a strong economic recovery is doubtful. Poor Rudolph (who will spend his adolescence being pulled in different and contrasting directions) is not going to find a strong empire when he reaches his majority.