Does string interpolation evaluate duplicated usage?

Yes, it will be evaluated twice. It can't know that it is the same value. For example:

Random rng = new Random();
Console.WriteLine($"{rng.Next(5)}, {rng.Next(5)}, {rng.Next(5)}");

It's exactly the same as if you just used the expression as a method argument - if you called:

Console.WriteLine("{0} {1}", person.GetFullName(), person.GetFullName());

... you'd expect GetFullName to be called twice then, wouldn't you?

If you only want to evaluate it once, do so beforehand:

var name = person.GetFullName();
emailBody = $"Good morning {name}, blah blah blah, {name} would you like to play a game?";

Or just use a normal string.Format call:

emailBody = string.Format(
    "Good morning {0}, blah blah blah, {0} would you like to play a game?",
    person.GetFullName());