Predicting school closures with historical weather prediction data?
I'm not aware of any datasets that are quite what you want (archived point forecasts for the next day).  There is less digestible data out there though.

NOAA NCDC archived messages
This website has archives of all messages that NOAA transmits.  The vast majority are unrelated to your needs, but if you look through the message types you might find something worthwile.

NCDC NOMADS
This site has archived GFS model grids.  This model is run 4 times per day and each run will have an analysis grid (current weather) and forecasts every 6 hours through the first few days of forecast time.


Most other archived data I could find are either observations or model reanalysis datasets, which won't work for your needs (except to better define the initial day t).  With the above datasets you could potentially put together the current weather at a specific place at 4 different times on day (t) and from each time have forecasts data for the next day, e.g.:

Observations / model analysis 12Z day t: 24 hour forecast for 12Z day t+1
Observations / model analysis 18Z day t: 18 hour forecast for 12Z, 24 hour forecast for 18Z day t+1
Observations / model analysis 00Z day t: 12 hour forecast for 12Z, 18 hour for 18Z and 24 hour for 00Z day t+1

From these you could then do your statistical analysis overall and explore how it may change as the day progresses and the model forecasts are shorter term.
If wading through this data is not something you want to deal with, an alternative is to start archiving forecast data yourself going forward and building up a database over time.
Some caveats...
The NOMADS GFS data is pure model data, no forecaster input.  This is not what decision makers will be using directly.  This data will be made more consumable and be tweaked by forecasters before dissemination to the public.  If the NOAA message archive contains suitable forecasts data, that data will have likely been filtered through a forecaster and close to what would be displayed on the NWS website as a forecast.
School closings are chaotic.  The same weather will not yeild the same closures.  Sometimes schools will stay open when faced with a worse forecast than a time they recently closed.  The criteria for closing may be a moving target as usable &quot;snow days&quot; are used up.  School closing thresholds will vary regionally (Florida schools would be closed Dec - March if subjected to a Michigan winter).  You may come up with a good forecast for when a school should close, but don't be surprised if it doesn't correlate well with actual closings.