The background for this question is that I know that many students starting to learn algebra focus on the standard construction of the quotient group $G/N$ instead of working with the universal property. Therefore it would be nice to give another construction, thus avoiding cosets, but so that the universal property is clear. This idea came to me years ago, but I've never found something.

For another example, there is a very nice and intuitive construction of the localization of a ring: $$S^{-1} A := \frac{A[\{X_s\}_{s \in S}]}{\sum_{s\in S} (s X_s - 1)}$$. The idea is: Invent new elements, and force them to be inverses to your elements of $S$. The universal property follows from the universal property of quotient and free algebra.

Perhaps quotient groups are a bit too elementary so that there could be another nice construction (please don't answer when you've just got this to say) ... anyway, any ideas are welcome :).

Sorry if this is too elementary for you ;-)