Windy Breeze

Okay, this is going to be a little rough since I'm not fluent in Japanese, but I have learned some things on an extensive level, so allow me to break it down...

"Windy" and "Breeze", while used in different situations, are still two words that have similar meanings. Therefore, putting them together in a regular sentence makes them kind of redundant.

"Kaze no soyokaze" is not a name, it's a phrase. It's the same as saying "The wind's breeze" or "The breeze of the wind". There is some redundancy since "Kaze", the word for wind, is used twice. But for a name, it's not what you want to go for.

"Soyokaze" by itself is the same as just saying "Breeze", since Soyo- is a modifying prefix for the root word, "Kaze".

And saying "Kaze Kaze" is the same as saying "Wind Wind". Redundant, but not quite the same as what you want.

THEREFORE! I have some suggestions for a name:

• "Soyono Kaze", two words. The first one word is written more like a name, the No at the end makes it slightly more formal. But over all, it's an informal way of saying "The breeze's wind". You might want to go for this since, as a name, it wouldn't sound like any typical Japanese name a human might have. And ponies don't usually have human names. However, if you really want something more natural-sounding in terms of names...
• "Soyono Kazane" is a name that could actually pass as a human's name in Japan. You still get the word play that could possibly be translated as "Windy Breeze" without being inaccurate. It just depends on what you want.

@Samurai Equine thanks for the suggestions. 

No problem. I could make more suggestions, perhaps better suggestions then that, but I wanted to stick with the examples you started with. 

Do you want me to write those names out in Japanese text for you?