What is the meaning of 'fly' in: "I wonder how many real amber mouth-pieces there are in London? Some people think that a fly in it is a sign."

Amber is a tree resin that has become fossilized. Occasionally, when the resin was still liquid an insect drowned in it and became trapped in the hardened amber. The presence of such an insect would prove that the amber was real and not an artificial substitute.

To give a bit more context: Holmes and Watson return, from a walk, to their rooms in Baker St.. The page-boy there tells them that a man called, then impatiently left. After talking with the page, Holmes says to Watson

Hullo! That's not your pipe on the table. 
He must have left his behind him.  A nice old brier
with a good long stem of what the tobacconists call
amber.  I wonder how many real amber mouthpieces there
are in London?  Some people think that a fly in it is
a sign.  Well, he must have been disturbed in his mind
to leave a pipe behind him which he evidently values
highly."

So not just any old mouthpiece but one made of amber. Sometimes a fly can get stuck in resin when that resin hardens and becomes amber.

Note the word "real". There are cheap pipes with fake amber mouthpieces, and more expensive pipes with real amber mouthpieces. Holmes sees the fly in this one and says "some people think" that the fly is a sign that the amber is real andthat the pipe is therefore expensive.

The meaning is that amber, which is fossilised tree sap, sometimes has small insects trapped inside it. A pipe mouthpiece made from a block of amber with a fly in it would, therefore be a mouthpiece with an ancient fly in it. As the workers making the mouthpieces would be working quickly to make as much money as possible they may well not have noticed the flies so a box of mouthpieces would, sometimes contain one with a fly in it.

These mouthpieces would be rare because most amber does not have flies so a myth could easily grow up that a mouthpiece with a fly inside was lucky, particularly as the mouthpieces would originally have been sold at random from the boxes in the tobbaconists' shops. The myth would be similar to the myth of the 'lucky' four-leaf clover (although four-leaf clover is probably more common than amber with flies in it).

The plot of the book and film Jurassic Park is based on the conceit that it would be possible to extract dinosaur blood from the bodies of ancient female mosquitoes which had fed on dinosaurs and then become trapped in tree sap which had then turned to amber over the millennia and that there would be enough lengths of DNA in the blood to re-create dinosaurs.