The HashSet<T> constructor has an overload that lets you pass in a custom IEqualityComparer<string>. There are a few of these defined for you already in the static StringComparer class, a few of which ignore case. For example:

var set = new HashSet<string>(StringComparer.OrdinalIgnoreCase);
set.Add("john");
Debug.Assert(set.Contains("JohN"));

You’ll have to make this change at the time of constructing the HashSet<T>. Once one exists, you can’t change the IEqualityComparer<T> it’s using.

Just so you know, by default (if you don’t pass in any IEqualityComparer<T> to the HashSet<T> constructor), it uses EqualityComparer<T>.Default instead.

Edit

The question appears to have changed after I posted my answer. If you have to do a case insensitive search in an existing case sensitive HashSet<string>, you will have to do a linear search:

set.Any(s => string.Equals(s, item, StringComparison.OrdinalIgnoreCase));

There’s no way around this.