First off you can get a new Safety button from Cole's in Maine and they are very easy to replace. There is a tab that sticks thru the top tang with a hole in it where a cross pin is inserted to retain the safety button. in the schematic for the safetyh button it appears to be a complete assembly which is the button and barrel selector tab so talk to Cole's and ask for confirmation on this. You will also want to get a new retaining pin for the safety button so you are starting with all new parts. Once your parts arrive you then remove the stock, remove the retaining pin, and put the new button in place and insert the retaining pin. After that check to see how it moves. If you find it's binding you will want a Diamond file so you can chuck the retaining pin in a Dremel and reduce the diameter at each tip of the retaining pin for a length of 0.10 inch until the safety moves with thumb pressure. Note, stuck safeties are a occasional problem with current Beretta's and I suspect your button got bent by someone trying to free it up with a hammer. The correct way to correct a stuck safety is to dress the ends of the retaining pin enough to allow the safety move freely.
Now for the second aspect of the safety that I sincerely hope is still in place. That is the spring for the safety button that also engages with that retaining pin. You may find the retaining pin in your gun has been ground down in length enough so it doesn't engage that spring and a new pin fitted to your action will result in a complete repair. However I suspect that a ham handed previous owner has been into your action and may have cut that spring. If so you will need to remove the trigger plate in order to replace that spring and that is
NOT an easy task for someone not familiar with this action. What I am saying here is if that spring is missing or cut you will need to take your gun to a gunsmith.
The page for the Safety, item 51. Note item 51 is at the top above the action and appears to be the complete assembly with the barrel selector. Item 50 is the retaining pin and only costs 6 dollars.
Beretta 680 Series part number 051 Safety
www.shop.colegun.com
The page for the safety spring, click on the schematic, the right button under the image and look for part 48 to see it's picture.
Safety Spring Beretta 680 Series Part number 048 C90014
www.shop.colegun.com
Note. The correct way to disable the auto safety on these guns is to remove the trigger plate and then remove the "lever" that connects the top lever to the safety. Many ham fisted owners will take a Dremel and grind off the tip that engages the safety. Note the safety spring is what retains the safety lever for the auto safe in the top tang of the action. This grinding can result in some damage to the safety spring and a lot of metal fines from the grinding in the action. So while you have the stock off your action look it over very carefully to see if you see any small metal shavings. If you do find shavings a trip to a gunsmith for a complete cleaning is in order.
Finally a diamond file is not at all exotic. Fact is that most hardware stores or sporting good stores will stock Diamond Files. They are called Knife Sharpeners. As for the need for a diamond file, those retaining pins are fully hardened and a file won't do anything and a grit stone will take nearing forever to turn the tips of the pins down. With a diamond file you do
NOT press hard with them, you use patience and a bit of water to do the work. Lean on them hard and all you will do is strip the diamonds off the file. I will also note that digital calipers are also an aid, on every Beretta I've have had with a stuck safety taking the diameter of the tips down by 0.005 inch was enough to get the safety moving easily.