How to correct spelling in Dictation on new iPad (3rd generation)
I found the answer from iMore's How to turn on and use Siri-like Dictation on your new iPad:


  If Dictation gets confused about any of the words you speak, it will underline them in blue on the screen. If the word is incorrect, tap it to get a popup menu with alternatives. Tap an alternative to switch to it. You can also simply edit voice dictation text like any other text.


Sure enough, single-tapping on Claire gives me an option to change the text to Clare, and of course it works other ambiguous words. 

A particularly useful example is when you mean Buy, and the dictation types By, with a blue underline. Tapping on that gives the two alternatives Buy and Bye.