Your question raises interesting ideas in the the areas of free will as well as the nature of consciousness. We might begin by assuming that humans have free will, as many philosophers do. It seems, after all, we are able to make choices, to "vary ourselves in some way without any externally sourced instructions". You seem to recognize that free will in a deterministic universe is a complicated issue, so I won't belabor the point here other than to point out some other good questions we have here on that topic:
What are the necessary conditions for an action to be regarded as a free choice?
Is free will reconcilable with a purely physical world?
What is the difference between free-will and randomness and or non-determinism?
The problem I want to highlight to you (which may be why you asked your question in the first place) is: Can we reconcile the notion of freedom while knowing that a robot we programmed — despite making choices we may not have known it would make — is acting in a way we intended (and perhaps even predicted) it to? If you program a robot to reprogram itself to make itself smarter and give itself its own goals, and it does that very thing, are those "choices" it made, or is it just following a set of rules/guidelines we defined? How many successful code-rewrites must occur before we say the robot is self-generating these actions? Can we ever say that?
This issue you run into here, it is no different from the issue of free will in humans. While our brains are made of more mushy materials, we do seem to exist in a very physical and determined universe. Our actions seem very influenced by our physical brain (if you damage it, we start acting differently, and altering the brain in more targeted ways with drugs seems to also have a profound affect). The influence which is perhaps most important to discuss here though (outside of direct external physical brain intervention) is the influence of the past. As we are raised, our brain is shaped not in some vague, ghostly way but physically in the neurons, and these physical changes (what account for our memories and consistency of self, why we see ourselves as the same self over time and not a new person at every moment) greatly influence how we behave and react to things. It is quite literally our own programming. We like to think we are "programmed" in such a way that allows us to "freely" make choices in our life, but do we?
There are many interpretations of free will (compatibilism, incompatibilism, pessimism, etc.) but if you want to hold a hard determinist view (an incompatibilist view, in contrast to libertarianism (philosophy)) view you will find it difficult to reconcile any notion of free will in robots or humans for the same reasons. In my answer to the first free-will-related question above, you will find my own proposal for a new definition of free will and how I reconcile the concept with human choice, and I leave it to others to explain how they do it under other views (as compatibilists, for example).
Regarding your comment:
Could a functioning A.I. system that can reprogram 'itself', (without causing any 'internal disfunction); could such a system
'reconfigure' various programs and information it is manipulating to
'come up with' a set of programs and/or info. that actually
contradicts certain ideas held by its programmers in such a way that
would prove the system was NOT acting even indirectly according to how
the programmers initially programmed it? In other words can an A.I.
system be 'programmed' and when 'used' it 'tells' the programmers some
of their fundamental ideas are wrong?
Can you? Can you act in a manner which does not accord to how you were raised and are genetically/biologically programmed? My gut tells me that nothing is random. A system can act in unpredictable ways, but those ways are not random, they are very caused, they are very determined by the previous inputs. So, sure, a system can act in a manner which contradicts its programmers only in that the programmers themselves had a poor notion of what their code would entail. Just as humans can seemingly act "out of character" too sometimes. The programmers thought that their algorithm would result in behavior somewhat like X, but it turned out like Y. The error is with the programmers, they predicted wrong; the system did not act on its inputs in a way that was random, or inherently unpredictable (just perhaps difficult to predict (obviously so for the programmers if they predicted something else)).