According to what I read in Wikipedia, lanyards were used originally to prevent from dropping the item they were attached to. One end of the lanyard would be clipped or tied to the body, such as a belt and the other end to the object. Old Webley revolvers had a lanyard that attached to the belt. A lot of marlin-spikes and other early marine knives had similar lanyards for the same reason.
It seems to me that most modern lanyards are much shorter. Presumably so they won't get tangled up in what ever the user is doing. I like them for their looks but I often wonder how useful a short lanyard is. Except for extracting the knife from where it sits in a sheath as Von Gruff pointed out.
I could see having a lanyard on an axe handle to prevent it slipping from your hand by wrapping the lanyard around the hand. On the other hand, an awkward swing of an axe that you can't let go of, might end up being very dangerous as well. So sixes to me as to whether a lanyard on an axe is a good idea.