It seems to me that it's common to think that words "Whether" and "Weather" share the same pronunciation, simple google search produces the articles like http://www.elearnenglishlanguage.com/blog/english-mistakes/weather-vs-whether/ that say:
The words weather and whether have nothing in common other than their pronunciation
On the other hand I learned from my favorite English Pronunciation Course which is called "Master Spoken English", what WH can be pronounced as HW: http://www.youtube.com/watch?v=iOnRC1VjTMw&t=1m10s But I heard from native speakers that it could be really odd if I start to say words like "When" with the initial H, like "Hwen", there is the whole episode of Family Guy around "cool whips" : https://www.youtube.com/watch?v=7ZmqJQ-nc_s
But, in the name of clarity, I think that it might be a good idea to say "Hwether". I am curious how weird it would be from a native speaker point of view?