Allowing a user to write a post from another website?

The idea here is to have a site (any site) which somehow has the user’s WordPress user-name and login (maybe as part of a sister site) to allow him to write a post on his own blog. The idea I have now is basically a simple form sent (perhaps with curl or socket) to the user’s blog admin_ajax.php, even though it is strictly not an AJAX call.

Are there any other methods which I could try or is there something built into WordPress itself already?

WordPress supports XML-RPC protocol for such things. That is how desktop blogging clients work.

Alternatively you can try to set it up via post by email.