Thomas Friedman is sick and tired of the Netanyahu government. He believes that Israel should have sided with the pro-democracy protesters, and that such a stance would have created a better relationship with the Egyptian people, the leaders of what Friedman calls "the new dynamic popular trend". In hindsight, it is true that Israel sided with a dead horse (Mubarak), but it isn't at all clear that it really matters.
First of all, events are still unfolding. We have no idea what kind of regime Egypt will now have. Will it be a liberal democracy, a flawed "illiberal democracy", a theocracy or a military dictatorship? Other than the unlikely first option, the three latter options are all equally possible. The protesters have won the battle, but not yet the war. If the military retains power, the fact that Israel stuck by the regime will be a positive thing in the eyes of the new leaders.
If democracy or theocracy await Egypt, anything Israel would have done regarding Mubarak would not help the relationship with the new regime. Egyptians hate us. Despite the cold peace between the two countries, Egyptian civil society boycotts Israel. Egyptian individuals who travel to Israel get punished by their professional associations. If Israel called on Mubarak to resign, Egyptians would see it as too little too late, and might even think of it as Zionist meddling.
As Friedman said, the Egyptian revolution had nothing to do with Israel. Similarly, Egyptian hatred of Israel has little to do with Israel's policy towards Egypt. The main problem is that they still see the Jewish State as a foreign colony in the Arab Middle East, as well as their anger at the way Israel treats Palestinians.
The only way for Israel to look more favorable in the eyes of Egyptians, and Arabs in general, is to reach a peace deal with the Palestinians. Even then, many would continue to hate us, but less.