You can access Object properties in two ways:
o.propertyname
//or
o.["propertyname"]
When using the bracket notation you have to put the propertyname in quotes or else it will be interpreted as a variable name (which in your case is exactly what you want). So in your case where you have stored the name of the property as a string, the way to go would be:
var variable = "propertyname";
o[variable];
/* /\ variable is replace with it's string representation "propertyname" */
You can even call methods this way:
var o = {};
var functionname = 'toString';
o[functionname]();
You can mix both notations, your example would look like:
var obj = 'objectName';
var prop = 'myID';
this[obj][prop]
// or this is possible too:
this.objectName[prop]