I vaguely remember hearing about a concept/theory/principle/statistic
that stated something like "every dollar put into a savings account at
a bank will be lent out nine times".

Is this, or something similar to this, an accepted (proven?) theory in
economics study? And if so, what is this principle called and where
can I find further academic references to its origin and current
application?

My guess is the statement is a bit dated, but essentually true. Nine
may be a low number considering many businesses barrow on their line
of credit 5 or more times per week and make partial repayments even
more often. Private parties who have automatic transfers from their
savings to checking account tend to push the number below 9.  Neil

I believe what you are referring to is the "Money Multiplier".  The
money multiplier is the amount of money a financial institution (bank)
can "create" when money is deposited with them.

This multiplying effect occurs because banks are only required to have
a small portion of their money in reserves... which means, if I
deposit $100,000 in my bank then they can put $10,000 in their
reserves (if the reserve rate is 10%) and then loan out the other
$90,000.  If whoever borrows that $90,000 consequently has $90,000 in
the bank then that bank can put $9,000 in their reserves and loan out
the other $81,000... and so on until eventually all these amounts are
loaned out:
$90,000 + $81,000 + $72,900 + 65,610 + $59,049 + $53,144 + $47,829...

The sum of all this potential "created" money is $900,000 (and the
origional depositor still has his $100,000 credited to his account). 
So from $100,000 the money multiplier has created a new $900,000 for a
total of $1,000,000.

I hope that's useful.  It brings me back to the good old days of
macroecon in college :)

Look up Money Multiplier and Reserve Rate on google and I'm sure
you'll get some good academic links.