A single word name for the "I don't know" gesture

The Gallic shrug includes both the moue (facial gesture) along with the shoulder shrug

"Raise your shoulders; hold up your hands, palms out; stick out your lower lip; raise your eyebrows;"

http://french.about.com/library/weekly/aa020901g.htm

The word for this gesture is pout. In the US and other anglophone nations, pouting usually indicates annoyance or exasperation. So, I wouldn't call this the "I-don't-know" gesture (words like shrug, etc, come to mind for this), but pout is the word you are looking for here.

pout: push one’s lips or one’s bottom lip forward as an expression of petulant annoyance or in order to make oneself look sexually attractive ^NOAD

to pout: may come close to what you are looking for:

To protrude the lips in an expression of displeasure or sulkiness.

To project or protrude.

Protruding lower lip out normally is accompanied with lifting of both eyebrows and symbolises informal and carefree gesture implying "no clue". It is not a polite or respectful gesture.