First, a few quick "whys":
1) The 7805 is actually a Motorola MC7805CP, date
code 7308 with gold leads. Very hard to exactly
replace.
Any reason why it would have to be an exact replacement?
In any case, the behaviour of the 7805 if you apply a
voltage to the output with the input floating may well
depend on the manufacturer and even the date (some
devices were improved over the years). Unless you have
a 1973-or-so data sheet from Motorola, I don't think
you know whether it will be damaged or not.
[...]
1) I have a 12 volt DC supply. 12 volts seems to be
within the VIN range for the 7805s whose data
sheets I've now read. Can I simply apply 12 volts?
Yes, but... The power disipated in the 7805 will increase, in
fact it will be more than doubled. To put it crudely, a linear
regulator acts like an automatic variable resistor. I have no
idea what current the load takes, let's call it I. If you supply
8V, then the power disipated in the 7805 is 3*I watts, if you
supply 12V it's 7*I. This may or may not be a problem.
2) Could I place a resistor in series between the 12V
supply and the 7805 to drop the voltage at the 7805
to somewhere around 8?
Yes. You need to know the maximum current the load will
draw, which will be much the same as the current drawn
from the PSU. Then just calculate the resistor to drop
4V at that current.
If you can find one, you could probably use a 7808 to supply
8V to the unit from a 12V supply. Or a 7805 'jacked up' with
a 3.3V zener diode (in series with the common lead to the
extra 7805 only).
My guess is that giving it 12V will be fine though.
What is the device, and do you have any idea how much
current it is going to draw?
-tony