What would BO5 be called? ?

In order for BO5{3-} to exist, the boron atom would need to have an oxidation state of 7+. That does not happen, and things that don't exist don't need names.
(NH4)3BO5 is fictitious.

BO₃³⁻ is called borate ion, but BO₃ does not exist.

I suppose you are asking for the name of BO₅³⁻ because each formula unit of the hypothetic compound (NH₄)₃BO₅ contains 3 NH₄⁺ ions and 1 BO₅³⁻ ion.

However, BO₅³⁻ ion does not exist.  This is because the maximum possible oxidation number of B is +3, but the oxidation number of B in BO₅³⁻ is +7.  BO₄³⁻ ion also does not exist because the oxidation number of B in this ion is +5.

 (NH₄)₃BO₅ 

That's the best I can do is write it correctly for you. 

Guess name:
Ammonium Oxyborate? 
(hypothetically)

Obviously this is 'Black Ops 5' - Cold War!!!
https://www.callofduty.com/blackopscoldwar

Oxyborate - borate + O2

Guessing...hyperborate.

I don't know