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It looks as if you are confusing cell range references and named ranges, which makes it hard to follow what you actually want. A1:INDIRECT(S1) is not a named range, it is a formula expression and would not magically appear in the named expression list. But you can name such an expression and use the name in your formula, just in the named range dialog enter the expression instead of a range.

You could also use a cell range reference where one corner cell is a named cell, such as A1:NamedCell where NamedCell previously was defined to, for example, A100.