What are some algorithms for finding the multiplicative order of an element in a group of integers mod m, besides the naive one?
If the prime factorization of $\lambda(m)\;$ or $\varphi(m)\;$ is known there are effective algorithms, see e.g.
Algorithm 1.4.3 in H. Cohen’s book A Course in Computational Algebraic Number Theory. A better available source maybe Algorithm 4.79: Determining the order of a group element from Applied Cryptography by A.J. Menezes et al. (you can download a pdf of ch. 4 from http://cacr.uwaterloo.ca/hac/).