You can't do that. The names of the properties of an anonymous type must be known at compile time. Why exactly do you need to do that?
You could achieve a similar effect by creating a sequence of dictionaries instead of anonymous objects:
string SomeConst = "OtherName";
var persons = GetPersons(); //returns list of Person
var q = persons.Select(p =>
new Dictionary<string, string>
{
{ SomeConst, p.Name }
});
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