As Separatix alluded to, this is an age old question which has not had an answer which satisfies everyone for thousands of years. You won't solve it in a few minutes.

The name I have most often seen associated with this problem is the Ship of Theseus. It was reported by Plutarch in his writings, before 150AD. Even then, it was already attributed to "Greek legend" suggesting it is far older than that:

The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.

From this, philosophers have drawn up lines with fancy terms such as endurable and perdurable to try to capture this conundrum along side an acceptable solution. For years, they have failed. It got even more difficult when science came along and started suggesting that the human mind could be encoded (which, by the way, is an assumption on your part, so I'd recommend touching on it in the story).

If you do start from the assumption that consciousness can actually be copied, there are still many options. My personal favorite is to suggest that, after the "copying" occurs, it is not so much that you have a copy of yourself as much as it is that your body is now twice as big, and in disjoint places. One of the lessons of the Ship of Theseus is that its very difficult to isolate a definitive self when engaging in such copying. Why not simply declare the "self" to consist of two bodies?

There's some precedence for this. Simple precedence can be found in the reattachment of a finger. We keep the severed finger on ice, but never one is it questioned that "this is the victim's finger." It's part of their "self." So having a body in two parts is not inherently forbidden. There's even really strange verbiages which have to arise when discussing organ transplants such as "he's using my lung."

There's also some really really interesting precedence in the world of conjoined twins. The sense of self associated with conjoined twins has always been complicated. For example, Krista and Tatiana Hogan are a fascinating case of twins conjoined in the brain. Impulses from one brain transmit directly to the other. However, there are times where their behavior is as though they are one individual.

Once you have this two bodied "self," obviously you will need to do something about it because the two bodies are likely to experience sufficiently different lives as to want to call them two "selves." This we also have a model for: divorce. In divorce, one takes a "unified body" and cleaves it in two, along with all of the property that body has attained. This process would have a natural corollary in the consciousness copying process. It even suggests a correct moral viewpoint for the clone killing problem. If the "self" agrees that one half of it should go away, who is to disagree. However, if the "self" is at odds with itself, the situation becomes less clear. Perhaps you have to send them to clone counseling, to come to an understanding of their greater self.

As Separatix alluded to, this is an age old question which has not had an answer which satisfies everyone for thousands of years. You won't solve it in a few minutes.

The name I have most often seen associated with this problem is the Ship of Theseus. It was reported by Plutarch in his writings, before 150AD. Even then, it was already attributed to "Greek legend" suggesting it is far older than that:

The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.

From this, philosophers have drawn up lines with fancy terms such as endurable and perdurable to try to capture this conundrum along side an acceptable solution. For years, they have failed. It got even more difficult when science came along and started suggesting that the human mind could be encoded (which, by the way, is an assumption on your part, so I'd recommend touching on it in the story).

If you do start from the assumption that consciousness can actually be copied, there are still many options. My personal favorite is to suggest that, after the "copying" occurs, it is not so much that you have a copy of yourself as much as it is that your body is now twice as big, and in disjoint places. One of the lessons of the Ship of Theseus is that its very difficult to isolate a definitive self when engaging in such copying. Why not simply declare the "self" to consist of two bodies?

There's some precedence for this. Simple precedence can be found in the reattachment of a finger. We keep the severed finger on ice, but never one is it questioned that "this is the victim's finger." It's part of their "self." So having a body in two parts is not inherently forbidden. There's even really strange verbiages which have to arise when discussing organ transplants such as "he's using my lung."

There's also some really really interesting precedence in the world of conjoined twins. The sense of self associated with conjoined twins has always been complicated. For example, Krista and Tatiana Hogan are a fascinating case of twins conjoined in the brain. Impulses from one brain transmit directly to the other. Because of this, there are times where their behavior is as though they are one individual.

Once you have this two bodied "self," obviously you will need to do something about it because the two bodies are likely to experience sufficiently different lives as to want to call them two "selves." This we also have a model for: divorce. In divorce, one takes a "unified body" and cleaves it in two, along with all of the property that body has attained. This process would have a natural corollary in the consciousness copying process. It even suggests a correct moral viewpoint for the clone killing problem. If the "self" agrees that one half of it should go away, who is to disagree. However, if the "self" is at odds with itself, the situation becomes less clear. Perhaps you have to send them to clone counseling, to come to an understanding of their greater self.