Setting anonymous type property name

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 }
});