The authors formulate a Herglotz-type variational principle on Lie algebroids, providing a common geometric language for several classes of dissipative mechanical systems. The resulting equations include familiar tangent-bundle, Lie-algebra and symmetry-reduced cases. Connections supply a global formulation that separates horizontal and vertical dynamics. The paper also derives energy-balance laws and analogues of Noether results in which dissipated invariants replace classical conserved quantities.