ORDER BY date and time BEFORE GROUP BY name in mysql

Another method:

SELECT * 
FROM (
    SELECT * FROM table_name
    ORDER BY date ASC, time ASC 
) AS sub
GROUP BY name

GROUP BY groups on the first matching result it hits. If that first matching hit happens to be the one you want then everything should work as expected.

I prefer this method as the subquery makes logical sense rather than peppering it with other conditions.

As I am not allowed to comment on user1908688's answer, here a hint for MariaDB users:

SELECT *
FROM (
     SELECT *
     ORDER BY date ASC, time ASC
     LIMIT 18446744073709551615
     ) AS sub
GROUP BY sub.name

https://mariadb.com/kb/en/mariadb/why-is-order-by-in-a-from-subquery-ignored/

I think this is what you are seeking :

SELECT name, min(date)
FROM myTable
GROUP BY name
ORDER BY min(date)

For the time, you have to make a mysql date via STR_TO_DATE :

STR_TO_DATE(date + ' ' + time, '%Y-%m-%d %h:%i:%s')

So :

SELECT name, min(STR_TO_DATE(date + ' ' + time, '%Y-%m-%d %h:%i:%s'))
FROM myTable
GROUP BY name
ORDER BY min(STR_TO_DATE(date + ' ' + time, '%Y-%m-%d %h:%i:%s'))