The Swiss-German spelling of Stäheli uses an umlaut "a" (ä) after the "t". In English the umlaut "a" (ä) is replaced with the letter combination "ae" to shift the sound of the pronunciation. This is explained in more detail on Wikipedia. I asked Marcel Muller at St. Gallen Staatsarchiv to pronounce the Staeheli name as it is said in Switzerland. He explained that it would sound like Schtaw-lee with emphasis on the first syllable.
Marcel also explained that Staeheli is a specific form of Smith (smith) which is defined as a craftsman or metal worker like a blacksmith or goldsmith. According to Ancestry.com Stäheli is the Swiss-German occupational name of an armorer. It is also the nickname for someone with a firm personality.
From the St. Gallen Staatsarchiv I have photocopies of the Burgerbuch for St. Gallen Canton published in 1901 describing the family names and genealogy for the region of St. Gallen. The history of Staeheli (Stahellini) is recorded to 1244. The book is in German so I will have a translation done. Meanwhile I will post the German information on this website.