The 9.3x62 is somewhat more powerful than the 9.3x57 and handles heavier bullets better. It's also more available and affordable in terms of brass and factory ammo. Personally, unless the rifle was particularly attractive in other ways, I would not buy a 9.3x57 with the intent to rechamber it to 9.3x62. I would buy a 9.3x62 in the first place.
Interestingly, I just picked up a Husky Model 146 (pre-war large ring FN Mauser action) in 9.3x57 with the intent of reboring and rechambering to .376 Steyr. It turns out that the land diameter of .375" bores is the same as the groove diameter of 9.3mm bores: .366". So basically the reboring cleans out the rifling from the 9.3mm bore, and then puts in the grooves to .375". This will be a test bed to see what the .376 Steyr can do with 300-grain bullets from a 24.5" barrel. It should be able to equal the .375 H&H with velocities in the 2500 fps range with less recoil, if my experience with it in the .376 Steyr Scout is any indication. I'm getting 2320 fps with my loads in the Scout's 19" barrel, so 5.5" more bore at, say, 25 fps per inch would suggest another 140 fps for 2460 fps, which is perfect, according to PH Kevin "Doctari" Robertson.
This particular conversion is worth it to me because no one currently makes a long-barreled rifle in .376 Steyr, so it's a custom proposition to begin with, and this particular rifle has a nice classic look to it. Others have done the conversion on small and large-ring 8x57 and 9x57 Mausers with few problems, so the 9.3x57 should work fine. And the fact that the rebore requires minimal metal removal should minimize the chance of distortion.