In this study, the temperature dependence of elastic constants $C_{11}$, $C_{33}$, $C_{rm E} = (C_{11}-C_{12})/2$, $C_{66}$ and $C_{44}$ of the iron-based superconductor Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ (0 $leqq x leqq$ 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 $Gamma_{1}$ (C4) symmetry. It suggests the existence of diverse fluctuations in this system. Gr{ u}neisen parameters were analyzed under some assumptions for structural and magnetic transition temperatures. Results showed that the Gr{ u}neisen 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. Gr{ u}neisen 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 Gr{ u}neisen parameters between $T_{rm S}$, $T_{rm N}$ and $T_{rm sc}$ shows $c$-axis elongation and its relevant role in the emergence of superconductivity in this system.
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