In The Devil's Arithmetic, what quick observation does the badchan make of Hannah?

The badchan’s “quick observation” that you speak of is incredibly insightful.  A badchan is a sort of hired comic or court jester who entertains at upper-class Jewish occasions.  As indicated in The Devil’s Arithmetic, he is a skinny man who is hired “to make up rhymes, sing songs ... [and] tell fortunes.” This particular badchan at the Jewish wedding feast spies Hannah and immediately sings this song. 

Pretty girl, with faraway eyes,Why do you...

The badchan’s “quick observation” that you speak of is incredibly insightful.  A badchan is a sort of hired comic or court jester who entertains at upper-class Jewish occasions.  As indicated in The Devil’s Arithmetic, he is a skinny man who is hired “to make up rhymes, sing songs ... [and] tell fortunes.” This particular badchan at the Jewish wedding feast spies Hannah and immediately sings this song. 

Pretty girl, with faraway eyes,
Why do you look with such surprise?
How did you get to be so wise,
Old girl in young-girl disguise.

This song is very significant.  Hannah is called “Chaya” when she goes back in time.  At this point, Hannah is the character of Chaya (who existed during the time of the Holocaust).  This is probably the reason why the badchan notices the “faraway eyes.”  Further, Hannah is surprised by almost everything she sees back in 1942.  In this case, Hannah is surprised to learn that arranged marriages are more common than marriages due to love.  This is the “surprise” of which the badchan speaks.  Further, the badchan notes that Hannah/Chaya is “wise” and hints at the switch of characters when he calls her the “old girl in young-girl disguise.”  The badchan’s next observation is about Hannah’s name.  Chaya means life:  “a strong name for a strange time.” This is significant for many reasons.  Hannah/Chaya eventually sacrifices herself for her friend named Rivka.  Through death, Chaya achieves life.  Also, the name is “strong” for this time period because so many Jewish people died that it is ironic that anyone Jewish would be named “life.”  The badchan recognizes, then, that Hannah/Chaya is a very special girl.