Wikipedia on Ottoman Empire gives its naming as coming from the Ottoman Turkish language, but on that very page, the name of the language is transliterated as Lisân-ı Osmânî. In Russian we call the empire and language "Османская"/-ий, also transliterating as "Osman[suffix]". The trail vanishes there... Were telephones that bad that the Turks said "Os" and the English heard "Otto"?
"Ottoman" comes from French. OED mentions that
Byron used the more correct form Othman...
We see that pronunciation with /th/ or /s/ is closer to the original, but in some languages (French, Italian) the pronunciation is different.
Far from the Turks having said "Os" and the English having heard "Otto", it is the variation in pronunciation coupled with a lack of exact equivalents in English that resulted in the curious spelling.
There are several s sounds in Turkish, not one, and each of these may not be pronounced identically by Turkish, Arab and Persian speakers.
Arie S. Issar & Mattanyah Zohar, in their book Climate Change: Environment and History of the Near East provide a brief insight:
The name "Ottoman" is derived from an 18th century "overcorrect" mispronunciation of the name of the founder, Osman, based on the Arab pronunciation 'Othman (with a sound like the English "th") but difficult for Persian and Turkish speakers who prefer the /s/ sound. (footnote at p.228)