Is it a good idea to use radios on mobile apps? If not, how would you improve the below design?
In this case I think it's ok to use radio buttons but be sure they are easy to tap.
If you have only 2 or 3 available choices the solution below could be nice. I also removed the "check" button since it adds a step that can be avoided to make the process simpler (you can show the user that you're checking while he's typing and show that the given name is valid/invalid).
You can add a message :
And to show that you're already checking (even if you're waiting they finished to type) you could use an animated icon (a loader in fact) :
Radios if list is small, drop down list if it is larger but also allowing to type in letters so it can be suggested from list.
Important, use large radios. But not too large. So people with bigger fingers can easly use it.
Look how this case solved in iOS7 dialog for changing ringtone
Radio's seem to be a good choice if switching is "common" and the list is not long, like in the case where a user might use the app and choose a different each time. Why? Because first of all a selectbox takes two touches rather than one, but on the other hand a radio like list takes far more space. I would also like to note that it would be good practice to not use actual radio buttons, but use bigger sized list items with a radio like element in front of them to signify selection (though do note that a checkmark icon seems to suggest multiple options being selectable).