I don't think anybody has developed that yet (at least I haven't seen it). The way to do this is will be by generating a cookie next besides the standard Mendix session cookies.
Theoretically you could establish this by setting the 'sessionTimeout' to a few days. (See custom environment settings)
I would not recommend this approach since will likely have some side effects.
Overriding the login behavior is possible, the LDAP module and the 'IP Range validator' both do this. They extend or replace the standard authentication method the platform uses. For the simplest example I would recommend taking a look a the IP Range validator the module hasn't been updated to 6 or 7 yet, but the concept and code in the latest releases is mostly the same.
This option does not give you access to the cookies though.
For remember me functionality on the client side you would have to built a custom cookie. There might be a few gaps in this design, but this is how I would start building this.
To auto-signin a second time
When you do this make sure to use a secure cookie validation. There is a semi working concept here on github, this does securely store and validate cookies. It uses JSON Web Tokens to store in the cookie and validate the response.
The cookie attribute should hold the secret and the token from JWT, that way you can guarantee that users can't mess with the token.
I know it sounds pretty abstract but I hope this would be helpful for some people.
If you publish your app in the appstore, it will remember the signed in user by default. The user needs to explicitly signoff if necessary. You can add pin security if you want to.
@Sid: It's probably not a good idea to attempt to do this yourself. As Jasper correctly points out this requires some work and thorough understanding of the concepts used.
A trial and error approach in development might be fine for some types of features, but defintely not for anything as security critical as user authentication.