I use a password manager now. It also generates random passwords based on rules you can tweak. They get stored offline and online so sync across devices. On my iPhone the app secures using touchid so is easy to use. It also has browser plugin If all your passwords are currently stored in your web browser then generally they can be extracted automatically into the password manager and it also check for dupes and advises which to change. I imagine most will recommend 1Password or LastPass with Keepass as the open source solution.
Your other option is to work out some scheme based on website name that allows you to mentally calculate what the password will be.
Note I keep any banking stuff completely separate in other encrypted means.