The only unofficial names for 3rd and 4th sons in use appear to be spare and (less commonly) spare to the spare. There are many online sources (both history and non-history) using spare for third, fourth, fifth etc sons, and in the UK these now include daughters.

Unexpected Heirs in Early Modern Europe: Potential Kings and Queens (Valerie Schutte, ed.) uses spare for King Henry III of France, who was the fourth son of Henry II of France, and also for Gaston, Duke of Orléans, third son (though 2nd surviving) of Henry IV of France.

The Historyextra site mentions ...plus at least one spare in case of accidents while spare to the spare is used with reference to Prince Edward, 4th son of Elizabeth II in this (not very historical) book.

Basically, Mark C. Wallace's comment is probably the correct answer but, as there seems to be some interest in this question...

The only unofficial names for 3rd and 4th sons in use appear to be spare and (less commonly) spare to the spare. There are many online sources (both 'history' and 'non-history') using spare for third, fourth, fifth etc sons, and in the UK these now include daughters.

Unexpected Heirs in Early Modern Europe: Potential Kings and Queens (Valerie Schutte, ed.) uses spare for King Henry III of France, who was the fourth son of Henry II of France, and also for Gaston, Duke of Orléans, third son (though 2nd surviving) of Henry IV of France.

The Historyextra site mentions ...plus at least one spare in case of accidents while spare to the spare is used with reference to Prince Edward, 4th son of Elizabeth II in this (not very historical) book.