I would "remove" Franz Josef I. With vastly different challenges if he were to succumb to an unfortunate accident in 1848, 1868 or 1889.
Removing him in 1848, ideally before he could abolish the brand new Austrian Constitution as it was drafted by parliament, would leave austrian throne to much more liberally minded Maxmilian, IRL mexican emperor. He would be only 16 and with a war on his hands, but if he got rid of PM Schwarzenberg, resisted his mother's influence and didn't repeal constitution, early liberal Austrian Empire would be possible. Though how to quell Hungarian revolution without Schwarzenberg negotiating Russian armies' aid... maybe giving in to their demands? Was it feasible? I don't even really know what Hungarians wanted. Could anyone enlighten me?
Offing him in 1868 would put Rudolf on the throne. IRL, he committed suicide in 1889 together with his mistress over Franz Josef's pressure to end their relationship. In 1868, he would be ten. With his father dead, obvious regent would be his mother, famous Sissi. She was quite liberal, so increasing her influence on state affairs right after the Ausgleich set a precedent how to deal with dissatisfied minority seems good. Maybe Slavs could get their Triple Monarchy? Austria-Hungary-Slavia? But she didn't get along very well with the rest of the court, and Sofie, Franz Josef's mother, was still alive. If she got on top, it wouldn't be pretty. Her tumor will get her in 1872, though. Not much time for her to screw things up.
His demise in 1889 would put Franz Ferdinand d'Este, IRL shot in Sarajevo in prelude to WW1, on the throne. He supported concessions to Slavs. Mainly to annoy Hungarians, but still, it was a good thing. He also wanted Austria to act as a great power, but not irrationaly - for example, he advocated caution when dealing with Serbia. He wasn't very likable though. But he was smart, pragmatic and energetic, and Austria desperately needed a dose of good leadership at that time, after political stagnation under Franz Josef.
Any of those would have been better than Franz Josef I., IMO. Maybe some liberal policies would bind minorities to the state, and Austrian Empire would survive the onset of nationalism. Man can dream