With regards to the "Remember Me" there are two areas in which this can be used however these are not specific API's.
The first is for the username and password on your login form this is not managed by Okta and we would not be able to provide the functionality unless using the widget as such you would need to implement your own code to generate a cookie to remember the users username this way when they next navigate to your login form the username is prepopulated. We would not recommend storing the password.
The second area is the "Remember Device" this is related to Multifactor and is managed by Okta. In order for this to work you will need to have multifactor policies defined in your org along with a sign on policy that prompts for factor Per Device or Per Session. This can be defined under Admin --> Security --> Authentication --> Sign On
If you have the correct policies in place and your login form handles multifactor challenges you will need to ensure you define a unique ID for the users device and send this through in each primary authentication request. The second compnent is to provide the user with an option to remember device and if selected send the following parameter in the verify call. ?rememberDevice=true /api/v1/authn/factors/:fid/verify?rememberDevice=true
We do provide documentation on the remember deive component which I have shared below, I would recommend reading through these articles and if you have any further questions related to this please feel free to log a support ticket as we can provide further assistance.