I am about to do the same thing to my STTNG. Everything I have read suggests you should replace both. However, I think you should follow the advice of whatever wiser heads speak here than I.

This game really needs its GI.

Always do both sides. It if you aren't experienced with PCB soldering, take it to a professional. It's really easy to damage the traces and it isn't expensive to have it done properly.

Yep replace both, unless you just like to do it over and over again.

Its MUCH easier if you use a desoldering station to remove the male header. But you can use one of those spring loaded vacuum ones but it will take a long time.

And be careful, as someone suggested, if you don't get ALL of the solder out, when you go to pull the pins out of the board you can damage the plated through holes. When that happens, some of the GI won't work and you'll have to fix that with a stitch or jumper.

Good time to convert your G.I. to LEDs to reduce power consumption and heat. The only problem with that is the game really needs an LEDOCD board for the G.I. with LEDs installed so that's another chunk of $$ to put into it, but well worth it IMO.

I have no affiliation with Herg or his products, just recommending it especially for games like STTNG that make so much use of local dimming of the GI. http://ledocd.com/cart/index.php?main_page=product_info&cPath=1&products_id=7

I have both the LED OCD and GI OCD boards in my STTNG and it makes the LEDs really work well.

You should always replace both sides of a burnt connector.

The plating on the pins has been compromised, and just replacing the female connector will cause it to burn again.

If you want to ship your board over to Columbus, I can fix it and ship it back. Won't take long!