Accurately understanding the propagation environment is a fundamental challenge in site-specific beamforming (SSBF). This paper proposes a novel generative SSBF (GenSSBF) solution, which represents a paradigm shift from conventional unstructured prediction to joint-structure modeling. First, considering the fundamental differences between beam generation and conventional image synthesis, a unified GenSSBF framework is proposed, which includes a site profile, a wireless prompting module, and a generator. Second, a beam-brainstorm (BBS) solution is proposed as an instantiation of this GenSSBF framework. Specifically, the site profile is configured by transforming channel data from spatial domain to a reversible latent space via discrete Fourier transform (DFT). To facilitate practical deployment, the wireless prompt is constructed from the reference signal received power (RSRP) measured using a small number of DFT-beams. Finally, the generator is developed using a customized conditional diffusion model. Rather than relying on a meticulously designed global codebook, BBS directly generates diverse and high-fidelity user-specific beams guided by the wireless prompts. Simulation results on accurate ray-tracing datasets demonstrate that BBS can achieve near-optimal beamforming gain while drastically reducing the beam sweeping overhead, even in low signal-to-noise ratio (SNR) environments.