Web app - how to replace ESRI branding with a custom graphic?

The "logo" option is documented as being a Boolean value that defaults to true. So if you put false, it won't show it; if you enter any value (or leave it off so it defaults to true), it will show their logo. (This is because JavaScript interprets any non-null value as true.) So you're left with the option to set it to false, and float your own logo over the map.