One representative time series signal from multiple time series data
Decision of representative is usually formulated as an optimization problem. The optimal representative minimizes sum of some dissimilarity measures from the observed signals. The average you indicated is optimal in the sense of minimization of Euclidean distance. If you want to minimize feet of perpendiculars, you can use the first principal component.

Now the problem is how to choose the dissimilarity measure. In signal processing, we often consider some probabilistic model and use the dissimilarity measure proportional to the negative log likelihood. This is often referred to as maximum likelihood estimation. For example, Euclidean distance corresponds to the negative log likelihood of mean parameter of Gaussian distribution. Even using the same Gaussian distribution, there can be various ways of modeling, and various dissimilarity measures are derived such as Mahalanobis distance and Itakura-Saito distance.