Indeed. While it is the secular surname of a pope-saint (Gregory VII), if you're seeking to honor him, why not just take the papal name Gregory?
I fail to understand what all the coyness is about, first teasing that a pope has been elected but not saying who he is, then releasing his chosen papal name but not disclosing his identity. It doesn't exactly inspire confidence.
I can reserve judgment about he "coyness", or the slow release of information, pending the revelation of more facts. I can see some reasons for drawing it out like this. Since they hardly have any kind of logistics to conduct even the most rudimentary of operations, if you truly believe that the guy they elected is Pope, you would to have everything ready before the flood of obligations comes in, so set up and define how your'e going to operate, etc. Even businesses take some time to plan out how their operations would work. In the current Vatican, when a new Conciliar gets elected, the existing curia tend to remain in place to serve at least as a "transition team", but absent such continuity, I can see that it would take some time to get everything set up, even if it's very simple and rudimentary to start. Not everyone can just go buy a cassock, continue living in his Mom's basement, and then have her make pancakes for you every morning before you begin your papal duties on the internet like "Pope" Michael did.
Perhaps also they need to get whoever it is consecrated a bishop or perhaps even ordained. Whether or not this elected is a bishop is one of the things I'm most curious about. If he's not a bishop, not sure how he can become the Bishop of Rome.
And, if you believe that your guy is legitimate, you might be concerned about the usurpers going after him and taking him out, so there may need to be some security apparatus put in place, or maybe he'll operate in quasi-hiding for some time and that location must be prepped.
So ... I have a "wait and see" about why they need to be so secretive and coy, as you put it.