Mary Jo Foley asks, Why ‘Azure’?  She uses Anand Iyer’s answer (from his post):

Azure is a vibrant, dynamic and uplifting color. Azure is also linked to the image of the blue sky and, by extension, ‘the cloud.’

I think there is a more direct association between azure and the Windows cloud:

1. Clouds are opaque. 
2. If you put a window in a cloud, you can see through to the clear sky.
3. The clear sky is azure.

So, Windows in the Cloud == Windows Azure