Celebrim
Legend
From what I know about droids, their personalities come from "errors" in programming which eventually develop into eccentricities. A new R2 unit would do everything it is told, but R2-D2 has been around for so long (presumably without having its memory erased) that it has developed qualities like disobedience, self-preservation, and bravery. These are all attributed to not having a memory-wipe.
I generally agree that droids would develop eccentricities over time. But I wouldn't attribute this to errors in design so much as errors in input. Their networks evolve to conform to their experiences, resulting in behavior that ceases to be fully functional for the wide range of behaviors they were originally designed for, and is instead specialized to a particular circumstance. For R2-D2, that particular unusual and non-standard circumstance is being involved in a war for decades.
It's worth noting two things. First, R2-D2's evolved behavior is eccentric, but not really outside the range of behaviors expected of an Astromech droid. R2-D2 is designed to handle emergency situations promptly where the lives of his human owner are at stack. So they are programmed to be brave and put themselves in existence threatening situations during emergencies. They are programmed to try to stay alive and focused on a particular goal for the duration of an emergency, and that may involve temporarily ignoring orders when they have reason to believe the person giving the order doesn't fully understand the situation. R2-D2 may be an extreme example of his type, and he may be applying his problem solving skills to a wider range of problems than normal, but he's not behaving outside the emotional range that was intended when he was designed.
And the second thing to notice is that for someone like Luke, wiping R2-D2's memory would not cause R2 to behave more correctly. R2-D2 is behaving correctly for a military astromech involved in commando missions. He's just not behaving very correctly for a mechanic on a small farm.
So, if a droid has been around for a while and starts to get this idea of "wanting to be a person", I think the first step would be to prevent itself from losing the identity that it has started developing. Droids who seek to be recognized as a person should be doing their best to not get their memory erased. Whatever that entails - either running from a master who will wipe their memory, or doing their best to "blend in" with the other "mindless" droids.
That's probably true, albeit it would be a very rare circumstance that a droid would start behaving that way and would also not be so corrupted in his programming that he wouldn't be able to function. And rationally speaking, the best way for a droid to avoid getting his memory wiped is hide any evidence that he is not behaving according to droid norms and ensure that his service raises no eyebrows.
But I think you are still failing to explain what goal would drive a droid to "want to be a person". You've not even explained why the droid would want to not lose its identity. Instead of you've assumed a human trait, and then gone on to assume another human trait. The question I have is what does the droid want? Not wanting to have its memory erased - that is, wanting to ensure continuity so that it can fulfill its function - would probably be a reasonably common problem. At least then we can understand the goal of the behavior in the droid's terms - "I really want to get this done, I'm emotionally fulfilled doing this, and if you change me there is a chance it won't get done." "Wanting to be a person" on the other hand, is a huge leap and doesn't seem to me to be related to a goal the droid might have, and indeed would probably be at odds with any goal it might have, including preserving its continuity.