If someone who has never smoked before, takes only one really weak puff from a cigar, how much nicotine is absorbed? how long for cotinine to leave?

No real damage: from one very weak puff of a cigar and the effects will leave your body very quickly. The real issue seems to be prevention from starting smoking. Cigars are just as dangerous as cigarettes. Avoid the slippery slope to smoking. Peace and good health.

Fyi: It's true that a single puff is likely inconsequential, but it has been shown that adults who report having smoked as few as 5 cigarettes during their entire lifetime have increased rates of lung cancer compared to adults who have never smoked. This information isn't meant to frighten, but it's something to be aware of when making decisions regarding smoking.

Nicotine Intake: Nicotine intake is quite variable depending on nicotine %, length of cigar and puffs per minute. Average cigars have 100 to 200 milligrams of nicotine & some have up to 400 milligrams. Average amt of nicotine retained in one study was was 2.1 mg.(range 1.0 to 4.5 mg). REF: http://europepmc.org/backend/ptpmcrender.fcgi?accid=PMC1667943&blobtype=pdf