This is an extended answer. An element of a lattice is called meet-irreducible if it cannot be expressed as a meet of two elements containing it. For a finite lattice, the meet irreducible elements are the unique minimal generating set for the lattice under meet. Your bases are exactly meet generating subsets of the subgroup lattice and the unique minimal basis is the set of meet irreducible subgroups (sometimes called primitive subgroups in the literature). It is known that the minimal degree of a faithful permutation representation of G is a lower bound for the size of this set. I am not sure what upper bounds are known.

You might want to look at Johnson, D. L. Minimal permutation representations of finite groups. Amer. J. Math. 93 (1971), 857–866.

This is an extended comment. An element of a lattice is called meet-irreducible if it cannot be expressed as a meet of two elements containing it. For a finite lattice, the meet irreducible elements are the unique minimal generating set for the lattice under meet. Your bases are exactly meet generating subsets of the subgroup lattice and the unique minimal basis is the set of meet irreducible subgroups (sometimes called primitive subgroups in the literature). It is known that the minimal degree of a faithful permutation representation of G is a lower bound for the size of this set. I am not sure what upper bounds are known.

You might want to look at Johnson, D. L. Minimal permutation representations of finite groups. Amer. J. Math. 93 (1971), 857–866.