When you capture a moga, a box will pop up and you can rename your moga to anything you want. But after that, you can't change the name anymore, if you want to change, go create another account and start naming the moga the name that you want.

EXTRA TIPS:

For all of you here, this question's previous answer is a little but wrong, here is the correct one.

Training Moga's Capture Rate: 100%

Begginer Moga's Capture Rate: 90%

Common Moga's Capture Rate: 59%

Uncommon Moga's Capture Rate: 39%

Rare Moga's Capture Rate: 29%

Super Rare Moga's Capture Rate: 19%

Epic Moga's Capture Rate: 9%

BEYOND EPIC Moga's Capture Rate: 4%

I don't think that answer is wrong, she/he never asked what the catching rates are, why would you put that, But you can't change it once you've named it, though it doesn't really matter, but if you really want the same moga with a different name, just capture another of the same one, now that they've upgraded monster galaxy, you can catch more of the same moga, for example, i have 2 Vuvus.

when you see your very first legendary pokemon, it will capture it with the capture rate of a master ball.

Kyurem has the usual Legendary capture rate of 3, the hardest to capture. Add to that the fact it is encountered at Lv75, it can be a difficult battle if you want to capture it.

Depends on the capture rate and resolution.

It's impossible without hacking.

If you hack it, the capture rate isn't that bad, so a Pokéball should work.