We explore certain extremals of the Sadowsky-type functional in Minkowski 3-space ${\mathbb{R}}_1^3$. Focusing on timelike rectifying strips, we provide a detailed characterization of the hyperelastic configurations. Specifically, we introduce timelike hyperelastic strips as surfaces derived from the solution curves of the minimization of this variational problem. Our investigation extends to the timelike planar critical points of the modified Sadowsky-type functional, where we demonstrate that these correspond to timelike hyperelastic curves situated on a timelike plane. Our research on timelike hyperelastic strips not only uncovers the fundamental Euler-Lagrange equations governing these structures but also highlights the geometric conservation laws that dictate their behavior. By deriving conserved quantities specific to these strips, we formulate the first and second conservation laws. This allows us to present significant insights into how these strips behave under both translational and rotational symmetries. We anticipate that these findings will pave the way for future studies exploring novel types of hyperelastic surfaces and their broader applications in geometric and physical contexts, particularly in relation to de Sitter and pseudo-hyperbolic spaces.