First off, you will need these tools.

Soldering Iron. Best if its mounted in a stand, with a tip cleaning sponge, or some other form of tip cleaner.
Solder spool (60/40 rosin core)
Flux (probably not required here, but handy to have.)
Solder wick or Solder sucker.
Diagonal cutters.
3.8mm game bit + driver.
CR2032 battery
CR2032 battery holder. (Harwin Inc. S8421-45R, available at Digikey or mouser.)
SNES game cartridge which needs its battery replaced. :)

Step 1: Taking the Game Cart Apart.

Remove the screws, and set them aside. When this is accomplished, remove the game cart pcb, and set the case aside.

If the battery is not yet dead, and there is a save still present, if you have a retrode, now is the time to make a backup of this save. :)

Step 2: Removing the Original Battery

If you have not already done so, plug in the soldering iron, and wait for it to heat up. When that is done, you may proceed to remove the battery.

The method I use to do so, is to heat up the smaller negative tab, and when it is hot enough, pull on the battery to release it. Repeat the process with the positive tab.

If you are lucky, the holes will be cleared of solder, and you can proceed to the next step. If not, that is where the wick or your solder sucker tool will come in handy. Get the holes cleared out.

Step 3: Modding the Battery Holder.

Now we have to work on the battery holder. I uncovered from the data sheet, that the inner edges from pad to pad, was exactly 18mm, the same distance as the original tabbed battery. Also, the height is just low enough to fit in an snes or super famicom game cart.

Since this holder is originally a surface mount part, we just have to make a few tweaks to make it a through-hole part, suitable for this application.

First off, we want to straighten both the positive and negative terminals.

Once we are done that, the next step is to clip a little bit off of the negative terminal using your diagonal cutters. (The positive does not need to be clipped, as it is 2.8mm wide. The negative unfortunately is 1.8mm wide, just 0.3mm too wide to insert as is.)

Step 4: Mounting the Battery Holder

Now, mount the battery holder in place, and solder it in place. If this is your first time soldering, it is advised you practice on something else, and read up on various soldering tutorials.

Step 5: Inserting New Battery and Putting the Game Cart Together.

Now, once the battery holder is soldered in, wait for it to cool down, then insert the CR2032 battery, so that the positive side faces up.

Then, put the game cart back together.

If you have previously made a backup of your save with a retrode, back in step 1, now is the time to restore it. :)