Why? The Subway is literally the only way I know to get around NYC. Yeah, technically I could use an Uber or (oh God) the bus, but the Subway is the most direct, critical means of navigating the city and actually compliments Uber in alot of ways.
I don't want to pick on you but alot of these insane ideas are just arising from governments being fucking impotent. The value of NYC's Subway system could probably be measured in something near a trillion dollars over the course of a year.
The city and the state need to stop fucking bullshitting and get together on how they're going to fix this thing. There is no real technological hurdle. There no hard decision to make. They already know what the correct decision is. They just need to get it done. Politicians need to stop bitching out and get their hands dirty. That's the problem. Not 24 hour service. Not the cost. Weak-kneed politicians.
/Rant
Considering the scope of repairs some form of inconvenience will have to occur. This would mean sections of the subway needing to be shutdown for non peak periods. Even with the funds you can't work while lines are running it's just flat out dangerous. So at a minimum a reduction in service hours for a period (1-5 years say) would at least air in some repair efforts.