If yours deletes your save and won't let you save anymore then chances are you will need to replace the battery.
This may work with other gameboy games but pokemon gold/silver are the only ones I know about.
Please Note: I take no responsibility for any damage to you, your gameboy, your game or anything else. My game and gameboy are fine and this shouldn't damage them, Just covering myself.
Please note: if you have ANY type of save that loads up then this will wreck that, otherwise lets get started
EDIT: quadcam24 has suggested in the comments that instead of using a screwdriver you could use fine tipped pliers to turn the security screw, this method would be easier and quicker than using a screwdriver if you could find pliers with thin enough tips
Step 1: You Will Need
A small straight head precision screwdriver,
A CR2025 battery, I'm sure other similar batteries e.g. CR2032 etc. will work but on the pcb it says CR2025 so thats what I used.
Some sticky tape,
You can also buy special batteries from the internet, which have tabs to solder directly to the pcb which will created the most secure connection possible, however if you tape the battery firmly then it should be just as good.
Step 2: Open It Up!
Step 3: Take Off the Old Battery and Put the New One On
Take the precision screwdriver and push it from where the tab is stuck onto the battery near the outside in towards the inside therefore removing the tab from the battery but keeping it attached to the pcb. Repeat on the tab stuck to the bottom of the battery.
Stick on the new battery with the tape making sure to get good contact between the battery with the tabs. If that comes loose then your save gets deleted :-(
Note that the battery goes in upside down (see the polarity markings on the pcb)
Please do not solder the tabs onto the battery as this is very dangerous and could cause the battery to explode
Step 4: Your Done!
Enjoy your game! :-)