Let's say someone already knows my Verizon FiOS router password and has previously hacked me and maybe my computers at home. If I change my router password, couldn't they see me doing it and know what I'm typing in, thus being able to continually hack me?
If the attacker has managed to either (a) modify the software running on the router, or (b) infect the computer used to change the password, then yes, the attacker will be able to steal the new password you set.
If you suspect that both the router and the computer has been hacked, I would recommend the following:
- Disconnect both the router and computer from the internet and each other.
- Do a factory reset on the router and do a complete wipe and reinstall of the OS on the computer.
- Connect the computer and router and reconfigure the router.
If your PC is infected with a key logger that sends information to the hacker, then yes he can see your password changes. A key logger can not only send what you typed in, but also screen shots.
You can use a PC that you know that it is not compromised by that hacker to change the password on the router.
Edit: based on your comments below I would do the following:
- Download the newest Firmware of your router from the official website of the router. Ask your internet service provider if you don't find it. You will need this software later on.
- Clean install your PC without being connected to the internet
- If you are on Windows or on Mac, install an antivirus program before connecting to the internet. (This means you should have the software ready on a memory stick.)
- Unplug your router from the internet.
- Factory reset your router with the reset button.
- Before connecting again to the internet: Install the new Firmware downloaded (point 1 above) and then change the default password of your router.
- Tell your sister (I suppose she is very young and hope nobody will be offended now) she should NEVER insert her password into a website that does not belong to that password. For example never insert your Facebook password in a website where the address bar shows a third party address. Always check in the address bar if the URL is correct. A small "spell mistake" is an indication that you are being hacked. (This is a very common hacking method. I still believe that her spam has nothing in common with your slow internet.)
- Observe your home network and PCs for the next couple of days or weeks.
I hope it helps. Let me know if you need further help or clarifications.