The mayor of Eilat, Meir Yitzchak Halevi, has announced he is going to file a petition with the Supreme Court against the lockdown. 

According to Yitzchak Halevi, Eilat is suffering from 75% unemployment due to the lockdown. he says Eilat is a green city, the most green city in all of Israel, but with its entire economy base don tourism, it has been destroyed by the lockdown.

Yitzchak Halevi wants to the government and Corona Cabinet to hold discussions about the situation in Eilat and come up with solutions to open things up. Halevi says, the government held 15 discussions about Uman before Rosh Hashana, and only one discussion has been held regarding the situation of Eilat. Yitzchak Halevi says that this city in Israel is collapsing and it has more residents than than the number of chassidim that travel to Uman

The Mayor of Eilat wants the city to be opened as a green island, with only one entrance open and visitors would need to bring a negative Corona test from within 48 hours to be allowed in. He has a complicated proposal of testing and rules that would apply in the city to allow tourists in, from within Israel and from outside Israel.

source: Ynet, haredim10, Kan News

I don't know how green Eilat actually is, but it is so separated from the rest of Israeli cities, distance-wise, that I do not see why Israel cannot easily isolate Eilat and grant it an exemption (if it is really so Corona-free). If, however, the plan is not good and a better plan cannot be devised, then forget it - ride the lockdown out like the rest of Israel.

I am not sure the Supreme Court is the way to go. I don't see them being helpful. Until now all petitions to the Supreme court for intervention in lockdown rules have been rejected, giving the government the authority to make such decisions. They would be better off finding ways to pressure the politicians that are relevant.