You can define a function to do that. You don't need to, though, just assign to MYOBJECT directly:
MYOBJECT.otherMethod = otherMethod;
function otherMethod(needThis){
    return needThis;
}
Functions are first-class objects in JavaScript, you can pass around references to them just like any other object. And you can create new properties on a JavaScript object at any time, just by assigning a value to the property.
If you want to define a function for it, the function will have to accept both the function and the name you want it to have on the object, like this, which looks like this:
var MYOBJECT = {
    // ...
    addMethodToObject: function(name, func) {
        this[name] = func;
    }
};
That's because there's no reason the property and the function have to have the same name, and there's no standard way to get the name of the function from the function instance. (There are a couple of non-standard ways, but nothing defined by the standard.)
The syntax I used there may be unfamiliar. In JavaScript, there are two ways to access properties: Dotted notation with a literal name, and bracketed notation with a string. Examples are the clearest way to demonstrate it:
var obj = {};
obj.foo = 42;    // Dotted notation with a literal name `foo`
obj["foo"] = 42; // Exactly the same thing, using bracketed notation and a string
var s = "foo";
obj[s] = 42;     // Again the same, because the string doesn't have to be a literal
...which is why this[name] = func; works; name is the string name of the property to add.