In this paper, we consider the computation of an eigenvalue and the corresponding eigenvector of a large sparse Hermitian positive-definite matrix using inexact inverse iteration with a fixed shift. For such problems, the large sparse linear systems arising at each iteration are often solved approximately by means of symmetrically preconditioned MINRES. We consider preconditioners based on the incomplete Cholesky factorization and derive a new tuned Cholesky preconditioner which shows considerable improvement over the standard preconditioner. This improvement is analysed using the convergence theory for MINRES. We also compare the spectral properties of the tuned preconditioned matrix with those of the standard preconditioned matrix. In particular, we provide both a perturbation result and an interlacing result, and these results show that the spectral properties of the tuned preconditioner are similar to those of the standard preconditioner. For Rayleigh quotient shifts, comparison is also made with a technique introduced by Simoncini & Eldén (2002, BIT, 42, 159–182) which involves changing the right-hand side of the inverse iteration step. Several numerical examples are given to illustrate the theory described in the paper.