In this study, the temperature dependence of elastic constants C_11, C_33, C_ E = (C_11-C_12)/2, C_66 and C_44 of the iron-based superconductor Ba(Fe_1-xCo_x)_2As_2 (0 ≦ x ≦ 0.245) have been measured. This system shows a large elastic softening in C_66 towards low temperatures. In addition to C_66, which originates from orthorhombic structural fluctuation, the samples near the optimal concentration show remarkable structural fluctuation in C_11 and C_33 elastic modes, which correspond to Γ_1

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... ) symmetry. It suggests the existence of diverse fluctuations in this system. Grüneisen parameters were analyzed under some assumptions for structural and magnetic transition temperatures. Results showed that the Grüneisen parameters for the inter-plane strain are remarkably enhanced toward the QCP, while those for the in-plane stress tend to turn down near the QCP. Grüneisen parameters for the superconducting transition are anisotropic and shows remarkable Co-concentration dependence, suggesting that the in-plane isotropic compression and inter-layer elongation enhance the superconductivity. The correlation of Grüneisen parameters between T_ S, T_ N and T_ sc shows c-axis elongation and its relevant role in the emergence of superconductivity in this system.