Though you can do this through PHP's time functions, let me introduce you to PHP's DateTime
class, which along with it's related classes, really should be in any PHP developer's toolkit.
// note this will set to today's current date since you are not specifying it in your passed parameter. This probably doesn't matter if you are just going to add time to it.
$datetime = DateTime::createFromFormat('g:i:s', $selectedTime);
$datetime->modify('+15 minutes');
echo $datetime->format('g:i:s');
Note that if what you are looking to do is basically provide a 12 or 24 hours clock functionality to which you can add/subtract time and don't actually care about the date, so you want to eliminate possible problems around daylights saving times changes an such I would recommend one of the following formats:
!g:i:s
12-hour format without leading zeroes on hour
!G:i:s
24-hour format without leading zeroes
!h:i:s
12-hour format with leading zeroes
!H:i:s
24-hour format with leading zeroes
Note the !
item in format. This would set date component to first day in Linux epoch (1-1-1970)