This is something that the citizens of Kazakhstan should decide, but to the extent anyone cares what an American who spent some time there and has a son with Kazakhstani citizenship thinks, I approve of replacing the "-stan" with a more Kazakh ending.
I think it would make Kazakhstan (or rather Kazakh Eli) more distinctive. In my experience, most non-Central Asians can't tell all the -stans apart. When I was there, a surprising number of acquaintances thought I was in Pakistan. To the extent that Kazakhstan wants to stand apart from the pack of post-Soviet Central Asian states (and also the South Asian -stans of Afghanistan and Pakistan), a name change would probably do a lot to help with that. It also would allow KZ to distance itself from "Borat." With the current name, it is impossible for any American publication to write an article about the country without bringing that film up.
In any case, the "-stan" suffix is originally Persian. Before the Turkic predecessors to the Kazakhs moved in, much of what is now Kazakhstan was part of greater Persia. Some Persian influences still remain and that is part of the country's history. But in the context of the government's overall project of Turkicicize (is that a word?) various place names, the change to Kazakh Eli does make some sense. (Although most name changes to date have been directed against Russian influences, not Persian. See, for example, Ust-Kamenogorsk/Oskemen, Aktyubinsk/Aktobe, and many other examples scattered around the country. Persian, at least, is viewed as more indigenous to Central Asia, even if it is not Kazakh.)
I think it would make Kazakhstan (or rather Kazakh Eli) more distinctive. In my experience, most non-Central Asians can't tell all the -stans apart. When I was there, a surprising number of acquaintances thought I was in Pakistan. To the extent that Kazakhstan wants to stand apart from the pack of post-Soviet Central Asian states (and also the South Asian -stans of Afghanistan and Pakistan), a name change would probably do a lot to help with that. It also would allow KZ to distance itself from "Borat." With the current name, it is impossible for any American publication to write an article about the country without bringing that film up.
In any case, the "-stan" suffix is originally Persian. Before the Turkic predecessors to the Kazakhs moved in, much of what is now Kazakhstan was part of greater Persia. Some Persian influences still remain and that is part of the country's history. But in the context of the government's overall project of Turkicicize (is that a word?) various place names, the change to Kazakh Eli does make some sense. (Although most name changes to date have been directed against Russian influences, not Persian. See, for example, Ust-Kamenogorsk/Oskemen, Aktyubinsk/Aktobe, and many other examples scattered around the country. Persian, at least, is viewed as more indigenous to Central Asia, even if it is not Kazakh.)