mailchimp: how to set a merge var value for a specific list member

In Mailchimp API v3 you can set a merge field value this way:

PUT {api_endpoint}/3.0/lists/{list_id}/members/{member_hash}

{
    "merge_fields": {
        "BIRTHDAY": "01/23"
    }
}

Where {member_hash} is md5 hash of member's email address.

You don't have to provide all merge fields. Mailchimp will update only those you've specified.