What does Hana's middle initial stand for?
It doesn't seem to be revealed anywhere officially, but based on fans' theory:
The "N" should be a word (that has similar pronunciation with) "Now".

Japanese fans found that all* characters in Hanayamata are related to Kamakura city in Kanagawa Prefecture, Japan.

From an anon user id 900RcZoqa on post no. 960 on 2channel,


  全部地名
  
  『ハナヤマタ』
  関谷なる → 鎌倉市関谷
  笹目ヤヤ → 鎌倉市笹目
  常盤マチ → 鎌倉市常盤
  西御門多美 → 鎌倉市西御門
  
  
  
  All of them are name of places
  
  "Hanayamata"
  Naru Sekiya → Kamakura city, Sekiya
  Yaya Sasame → Kamakura city, Sasame
  Machi Tokiwa → Kamakura city, Tokiwa
  Tami Nishimikado → Kamakura city, Nishimikado  


that leaves Hana N. Fountainstand.

Turns out there's also a place in Kamakura city called Imaizumidai. Imaizumidai in Japanese is 今泉台. If each character is translated to English,


今 = Now
泉 = Fountain
台 = Stand


they match with "N. Fountainstand".


  ハナ・N・フォンテーンスタンド → 鎌倉市今泉台（Now→今 fountain→泉 stand→台）


As for what does the "N" stand for, we can't really know, since as it's been mentioned, フォンテーンスタンド might be read as Fountainstand, Fontaine-Stand, Fonteyn-Stand, Founteyne-Stand, etc. Thus, the "N" might be Now, Nau, Naw, etc.



*Except Sachiko Yamanoshita. There's no Yama-no-shita in Kamakura city, but there's Yama-no-uchi and Saka-no-shita. (From Japanese Yahoo! Answers)

Also, there's someone who travelled around Kamakura to find these places. (1, 2)