One of the most attractive distance measures between two curves is the
Fréchet distance, which is the smallest leash length between a dog on one curve and
its owner on the other.
Algorithms for computing it have been studied since the mid-90's, perhaps starting with
H. Alt and M. Godau. Computing the Fréchet distance between two polygonal curves. Intl. J. Computational Geometry and Applications, 5:75-91, 1995.
[Image from Wouter Meulemans' web page.]
Once you have committed to this distance measure, it is natural to define a median curve as that which
minimizes the maximum Fréchet distance between it and the curves in your collection.
And indeed this has just been explored in a recent Ms. thesis:
Benjamin Raichel and Sariel Har-Peled. "The Frechet Distance Revisited and Extended." 2012.
(conference paper link).
The exact median curve of $k$ $n$-vertex polygonal chains can be computed in $O(n^k)$ time.
But under a natural restriction that the curves are ``$c$-packed," the exponential time complexity is reduced to $O(n \log n)$ for a $(1+\epsilon)$-approximation. All of this is detailed in Raichel's thesis.
I doubt there are existing implementations (because this is so new), but examining this
literature should at the least provide you with one natural model of an "average" curve.