Can I add arbitrary properties to DOM objects?

Yes, you can add your own properties to DOM objects, but remember to take care to avoid naming collisions.

document.getElementById("myElement").myProperty = "my value";

HTML5 introduced a valid way of attaching data to elements via the markup - using the data- attribute prefix. You can use this method in HTML 4 documents with no issues too, but they will not validate:

<div id="myElement" data-myproperty="my value"></div>

Which you can access via JavaScript using getAttribute():

document.getElementById("myElement").getAttribute("data-myproperty");

Do you want to add properties to the object, or attributes to the element?

You can add attributes using setAttribute

var el = document.getElementById('myelement');
el.setAttribute('custom', 'value');

or you can just add properties to the javascript object:

var el = document.getElementById('myelement');
el.myProperty = 'myValue';