Spin-phonon coupling, the interaction of spins with surrounding lattice is a key parameter to understand the underlying physics of multiferroics and engineer their magnetization dynamics. Elementary excitations in multiferroic materials are strongly influenced by spin-phonon interaction, making Raman spectroscopy a unique tool to probe these coupling(s). Recently, it has been suggested that the dielectric and magnetic properties of 15R-type hexagonal BaMnO3 are correlated through the spin-lattice coupling. Here, we report the observation of an extensive renormalization of the Raman spectrum of 15R-BaMnO3 at 230 K, 280 K, and 330 K. Magnetic measurements reveal the presence of a long-range and a short-range magnetic ordering in 15R-BaMnO3 at 230 K and 330 K, respectively. The Raman spectrum shows the appearance of new Raman modes in the magnetically ordered phases. Furthermore, an additional Raman phonon appears below ~ 280 K, possibly arising from a local latticedistortion due to the displacement of Mn-ions, that exhibits anomalous shift with temperature. The origin of the observed renormalization and phonon anomalies in Raman spectra are discussed based on the evidences from temperature-and magnetic-field-dependent Raman spectra, temperature-dependent x-ray diffraction, magnetization, and specific heat measurements. Our results indicate the presence of magnetostriction and spin-phonon coupling in 15R-BaMnO3 thus suggesting that the optical phonons are strongly correlated to its magnetoelectric properties.