I think the answer to the original question is that there is no special name for the involution (otherwise it would have turned up by now). My first encounter with it was in the 1979 Kazhdan-Lusztig paper on Hecke algebras, where they use a bar notation and combine this involution on Laurent polynomials with the inversion in a given Coxeter group to get an action on the Hecke algebra of that Coxeter group. The bar notation makes it unnecessary to invent a name for the involution on Laurent polynomials, but "bar involution" will certainly do.

By the way, a ring of Laurent polynomials (say over $\mathbb{Z}$) provides a nice nontrivial example for a graduate algebra course, even though it rarely if ever occurs in textbooks. It's natural to ask what are the prime ideals and factor rings, etc. Most often the examples of commutative rings which students see are too boring and predictable to motivate the ideal machinery.