The Neighborhood Council system was set up precisely to diffuse the power that created it. That is why they have no actual authority, that is why the voting rules make it impossible to coordinate NC elections with City elections. The Council had to do something because there was a lot of grumbling, so they came up with a way to tie up activists with meetings and debates and elections of no import.
The Mayor and Council will no way voluntarily cede any more power or influence to the NCs. It's not in their interest.
What is needed is a petition drive to increase the number of council districts to 96 or 99, matching the borders of the NCs (or not if they are not of near enough population size). The language would be easy to draft-- just change a couple of lines in the Charter.
(Does that sound like too many? Not at all. Many smaller cities have much bigger councils than we do. Having "local" districts as large as ours distorts democracy-- money plays too big of a role.)
The new Council would have to better reflect the wishes of their constituents-- they would be easier to run against, but of course they will have a conflict of interest then, too, and also not want to give more power to the NCs until sufficient force is brought to bear. But it would be easier.