Well, I'm not 100% sure about the actual pronunciation of "Weiner" in this particular case, but if it works at all like standard German, the pronunciation of this spelling (amerikanische Art) is "Why-ner".
The classic hot-dog type pronunciation and spelling is "Whee-ner" and "Wiener", respectively.
For some unfathomable reason, NE American Jewish families mostly reverse the original German pronunciation of the 'ei' and 'ie' in their names, as mocked in the classic 'Young Frankenstein,' a shift the non-Jewish German immigrants generally didn't make. One supposes this goes back to an ethnic differencing but I won't hazard a guess as to how it started or by which group, though it does seem to be the Jewish part which gets excited about it.
This particular dipshit's name is therefore usually pronounced 'Weener' in the media, like the English pronunciation of the sausage which is actually spelled the other way. The name of the sausage, 'Wiener,' comes from its origin in the Austrian city of Vienna ("Wien" in German) and of course the analogy to the similar size, shape and color of Mr. Happy is pretty obvious.
In its original German context, his name would sound to us like 'Viner' and means 'Wine maker.'