Strongly stable ideals are a class of monomial ideals which correspond to generic initial ideals in characteristic zero and can be described completely by their Borel generators, a subset of the minimal monomial generators of the ideal. Francisco, Mermin, and Schweig developed formulas for the Hilbert series and Betti numbers of strongly stable ideals in terms of their Borel generators. In this work, a specialization of strongly stable ideals is presented which further restricts the subset of relevant generators. A choice of weight vector $w\in\mathbb{N}_{> 0}^n$ restricts the set of strongly stable ideals to a subset designated as $w$-stable ideals. This restriction further compresses the Borel generators to a subset termed the weighted Borel generators of the ideal. A new Macaulay2 package wStableIdeals.m2 has been developed alongside this paper and segments of code support computations within.