My preferred spelling of Turretin's name is "Turretin," with two total t's. There are, however, several variants:
1. Turrettin: this is by far the most common variant, perhaps even more common than my preferred spelling.
2. Turrettini: this is an attempt to return the family name to its original Italian.
4. Turrettinus / Turretinus: these are attempts to Latinize the family name.
5. Turretine: this is an odd (but surprisingly popular) variant that seems to be based phonetically on a mispronunciation.
6. Turtin: this is a rather crude phonetic variant.
One occasionally sees one of 1-4 with only one "r," but this is probably just a spelling error. Also, occasionally - due to older typefacing - one sees a "v" for the "u"('s), particular in the Latinized forms.
There's also some variation as to the first name, again my first choice is first, followed by the most to least common variants.
2. François / Francois