I'm seeing more and more stores selling huge packages (24/32/48) packs
of AA and AAA batteries, and usually not a name brand like duracell or
energizer.

How long does it take for you to use that many batteries? I'd be
worried that the shelf life expires or they leak before I get around
to using them.

So it's a crap shoot knowing how many milli-amp hours of capacity these
batteries have.

I have a gut feeling that the amount of juice in these things is
proportional to weight.

I've never seen a completely dead battery that crumbles to dust
(leaks and corrodes the guts of your device is something different)
or can be blown away with a sneeze, not even those tiny coin cells.

The capacity *when new* may be roughly proportional to the weight
of the battery (and the active chemicals in it) when new, comparing
only those that use the same type of chemical system (e.g. carbon-zinc,
alkaline, NiMH, Lithium-ion, Nickel-Cadmium, lead-acid, etc). Don't
count the weight of terminals, labels, plastic holders, etc.
Rechargeable batteries generally have labels with a milli-amp-hour
rating (whether that rating is accurate is another issue.)

So D cells have more ampere-hours than AA cells which have more
ampere-hours than AAA cells, provided nobody cheats and fills space
inside them with air. (There are some D-cell adapters that hold 3
AAA cells in parallel. Much of the space is plastic and air. You
may get the same voltage when fresh but it's going to wear out much
faster than a real D with the same chemical system.) Car batteries
are large because they need to be to give enough starting energy
to crank the engine, especially in cold weather.

This does *NOT* mean that a half-dead battery weighs half of a fresh
one. Aren't most types of batteries supposed to be sealed (lead-acid
being an exception for some car batteries)?

There is only a limited amount of space in an AA cell, and trying
to cram in more after it's full (to get more energy out) isn't going
to work. So all AA cells are going to weigh about the same, except
for the cheaters who fill part of the space with air.

Are there any studies or has anyone tried to correlate the weight (mass)
of a brand new AA or AAA battery with how many watt-hours can be had
from them?

There is the formula E = mc**2. However, don't count on being able
to measure the difference between the weight of a fully-charged
battery and a dead one. Even an atomic bomb uses only a tiny
fraction of the mass-energy of the active material (plutonium)
although you may have trouble finding all the remaining pieces to
weigh them.