Burned Out Tantalum Capacitor Replacements
It looks like your fried cap is connected right next to the input terminals of the +12&nbsp;V rail and GND. Here are the steps I would try to get the drive running again:


Remove the cap. Power the hard drive, chances are it will work
anyway because the cap just acts as a small filter for the incoming
+12&nbsp;V rail and the slightly increased ripple voltage on this rail will likely be tolerated by the rest of the hard drive. This
test will also tell you if anything else has failed along with the
cap or if you're lucky and the cap is the only bad part.
If step 1 was unsuccessful, you can put pretty much any cap with at
least the same capacitance and at least the same voltage in place of
the fried cap. Maybe the drive will work now? If you just need the
drive to recover the data, any half-decent electrolytic (including
tantalum) capacitor will likely do the job long enough until you
finally scrap the drive. Don't worry too much about the ESR. As the cap appears to be just in parallel to the 12&nbsp;V input, it's not worth counting every milliohm. However, your're of course right: The smaller the ESR, the better.
However, by the looks of it, the drive is so old that you value not
only the data but the cool vintage hardware itself. Here's what I
would do for a good, permanent repair: Sizes of SMD tantalum
capacitors are standardized. Yours
appears to be one of the bigger ones (C?, D?). Try finding a
replacement cap with the same size and capacitance, and you're very
right about the voltage - 25 V or more is a very good idea. There is
a huge derating to be considered when using tantalum capacitors, and
using a cap with twice the voltage it is subjected to in your
application is not a bad idea at all. One last hint: While aluminium
electrolytic caps have the mark ("bar") on the negative end,
tantalum capacitors are marked on the "+" end!
As we're aleready talking about derating and fried tantalum caps: Besides (even very short) voltage spikes causing tantalum caps to fail catastrophically, they are also very sensitive to current spikes (as they will appear when you hot-plug the power connector into the drive). If you have the space, using an aluminium electrolytic cap as a replacement will be more robust. If you want to go for a "good as original" vintage repair, try to stay away from hot-plugging your drive.


Example random size chart (it gets more confusing when you consider not only the length and the width, but also the height):



Related: https://electronics.stackexchange.com/a/80458/930