A well-known result of Y. Colin de Verdière states that given any compact connected manifold $M$, with $\dim M\geq 3$, and any finite sequence $0\leq a_1\leq\dots\leq a_k$, there exists a Riemannian metric on $M$ such that the first eigenvalues in the spectrum of its Laplacian are $0\leq a_1\leq\dots\leq a_k$. The original paper can be found here. Therefore, there is no hope for a "universal" lower bound on $\lambda_1(M)$.

A well-known result of Y. Colin de Verdière states that given any compact connected manifold $M$, with $\dim M\geq 3$, and any finite sequence $0<a_1\leq\dots\leq a_k$, there exists a Riemannian metric on $M$ such that the first eigenvalues in the spectrum of its Laplacian are $0< a_1\leq\dots\leq a_k$. The original paper can be found here. Therefore, there is no hope for a "universal" lower bound on $\lambda_1(M)$.