Firstly, you can use data
attributes to identify your buttons more reliably:
<button class="itemButton" data-button="A">
Button A
<span class="itemCount">0</span>
</button>
<button class="itemButton" data-button="B">
Button B
<span class="itemCount">0</span>
</button>
<button class="itemButton" data-button="C">
Button C
<span class="itemCount">0</span>
</button>
Then you can use this to identify each item in the object. Note, you need to use an array index notation when accessing an object property using a variable, and also your object syntax was incorrect.
var items = {
"A": { timesClicked: 0 },
"B": { timesClicked: 0 },
"C": { timesClicked: 0 }
}
$('.itemButton').click(function () {
var button = $(this).data('button');
items[button].timesClicked++;
$(this).children(".itemCount").text(items[button].timesClicked);
});