That's an interesting request. I know it can be set to be 00:00 hours, so that is midnight.
In my experience this is likely to be an issue, given that most institutions do not have support available at this time of the day. I always suggest that business hours is a far better time, just in case your users need support when submitting.
Of course, that's just my opinion. I'd like to hear others.
I do understand the problem with 00:00 as this then requires the date to be set to the next day, sometimes leading to confusion among students.
However, that is the accepted way the 24 hour clock works and so allowing anything different could also cause confusion - is 'midnight' on 21st June very early on the 21st or very late on the 20th?
Why Tom? You want to encourage staying up late to submit?
You can set all datetime selectors in any form of moodle to be with 1 minute accurasy. I think this is the best solution as you will be able to set deadlines at 23:59.
You can change function MoodleQuickForm_date_time_selector, file lib/form/datetimeselector.php. Change 'step' parameter from '5' to '1' as following.
$this->_options = array('startyear' => $calendartype->get_min_year(), 'stopyear' => $calendartype->get_max_year(),
'defaulttime' => 0, 'timezone' => 99, 'step' => 1, 'optional' => false);