Starting from two low-cost, low-environmental-impact polymers belonging to the Polyethylene Glycol (PEG) family, i.e., PEG 800 and PEG 1000, two form-stable phase change materials were produced. The two PEGs differ in molecular weight and, as a consequence, the melting and crystallization range of temperatures. The PCMs were obtained, including the PEG, in a liquid state, inside the pores of Lecce Stone flakes, obtained as waste pieces from its processing. A simple and inexpensive impregnation process was selected to produce the PCMs, thus adopting low-environmental-impact materials and cheap processes, and respecting circular economy principles. The two PCMs, the first composed of PEG 800, namely LS/PEG800, and the second composed of a 50/50%wt. mix of the different LS/PEGs, i.e., LS/PEG800_LS/PEG1000, were added as aggregates to four types of mortars, based on aerial and hydraulic lime, gypsum, and cement. The obtained mortars were characterized in their fresh state to assess their workability, and in a solid state after a proper cure to determine their characteristic Latent Heat Thermal Energy Storage (LHTES) properties and mechanical properties in both flexural and compressive modes, taking the mortars not containing any PCM as the reference. The results revealed that, with the proper selection of mortar formulations, it was possible to achieve suitable workability and adequate mechanical characteristics. The selection of a PEG with a low range of phase change temperatures, such as PEG 800, allows one to obtain mortars characterized by a melting/crystallization range that can be considered appropriate in applications characterized by cold climates. The production of a mixed PCM, composed of both PEGs, led to mortars displaying a large interval of melting/crystallization temperatures, which could be suitable in both warm and cold climates.