We develop a generalized holographic dark energy model based on the Rényi entropy, which introduces a logarithmic deformation of the Bekenstein--Hawking entropy and is characterized by a non-extensivity parameter $\alpha$. By adopting the future event horizon as the infrared cutoff, we formulate the New R\'enyi Holographic Dark Energy (NRHDE) scenario and derive a modified holographic energy density that reduces smoothly to the standard HDE limit for $\alpha\to0$. Starting from the Rényi entropy formalism, we obtain a closed and self-consistent set of evolution equations for the dark energy density parameter $\Omega_d$, the equation-of-state parameter $w_d$, and the deceleration parameter $q$. We perform a detailed numerical investigation of the background dynamics over a physically reasonable range of the holographic parameter $c$ and the R\'enyi deformation parameter $\alpha$, and show that the NRHDE model predicts a late-time phantom regime over an extended region of the $(c,\alpha)$ parameter space, with a smooth approach toward the cosmological-constant boundary $w_d=-1$ as either parameter increases. We further provide a global characterization of the parameter space by means of two-dimensional maps of the present-day equation-of-state parameter and the transition redshift, which clarify the joint impact of $(c,\alpha)$ on the late-time cosmological evolution. Finally, a qualitative comparison between the NRHDE background predictions and observational Hubble data from cosmic chronometers is presented as a consistency check of the model at the background level. The NRHDE framework, therefore, constitutes a minimal and thermodynamically motivated extension of holographic dark energy, offering a flexible platform for future quantitative tests with late-time expansion data.
