I have taken a look at the code, here is what we can do:

• we will need to add ActivityConfig, to use an option button for 12 hours system and 24 hours system.
• we can use the same datasets, as the difference between the 2 systems is only in the displayed hour, and after checking the datasets, the hour isn't controlled via dataset, as in each sublevel, differentTargetH( ) is called.
• we can change differentTargetH( ) to be as following to support both systems:

  function differentTargetH() {
    items.targetH = Math.floor(Math.random() * 24);
    while(pastQuestionsH.indexOf(items.targetH) != -1) {
        items.targetH = Math.floor(Math.random() * 24);
    }
    pastQuestionsH.push(items.targetH);
    if ( items.twelveSystem ) displayTwelveSysetm();
}

• differentTargetH( ) is now supporting both systems, we will add a new function to convert the items.targetH form 24-hour system to 12-hour system if it is 12-hour system, else we won't convert it and we will display the items.targetH as it is

function displayTwelveSystem() {
        if( items.targetH > 12 ) {
            items.targetH -= 12;
        }
}

• Nothing will be changed in differentCurrentH( ) as we aim that items.currentH has a different value from items.targetH

• for checkAnswer(), we need to call displayTwelveSystem in case of 24-hour system so that we can check if items.targetH equals to items.currentH.

I think nothing else will be modified, to support both systems.
what do you think ?

This looks like a good way to go, thanks!

After checking the result, I have some doubts on the usefulness of displaying AM or PM.
As we display hours from 00:00-12:00 on the clock, what is the interest specifying we want 4PM or 4AM?

Hi,
You are right. Nothing on a pedagogic point of view. AM and PM would only be interesting if we would treat duration type of problems. This is not the case here.