Converting XmlNodeList to List<string>

Yes, it's possible using LINQ:

var memberNames = membersIdList.Cast<XmlNode>()
                               .Select(node => node.InnerText)
                               .Select(value => int.Parse(value))
                               .Select(id => library.GetMemberName(id))
                               .ToList();

Cast<XmlNode>() call is necessary, because XmlNodeList does not implement generic IEnumerable<T>, so you have to explicitly convert it to generic collection from non-generic IEnumerable.

And yes, you can merge all Select calls into one if you want:

var memberNames = membersIdList.Cast<XmlNode>()
                               .Select(x => library.GetMemberName(int.Parse(x.InnerText)))
                               .ToList();

Why don't you use LINQ to XML ?

List<string> memberNames = XDocument.Load("path")
                           .XPathSelectElements("//SqlCheckBoxList/value")
                           .Select(x => library.GetMemberName((int)x))
                           .ToList();