When I think of weapons made of ice, I usually think of Stahlrim weapons from Skyrim, do the ice weapons from the show have a name? What about the books?
It's hard to prove a negative but as far as I am aware no formal name is ever given to the ice weapons. When asked what they are made out of George R. R. Martin answered the below, if he was ever going to formally name them that would seem like the perfect opportunity.
Shaw: Do you know what substance an Other sword is made from.
Martin: Ice. But not like regular old ice. The Others can do things with ice that we can't imagine and make substances of it.
As @Aegon has mentioned in a comment we do see the Others communicating in the Prologue for A Game of Thrones so it's likely that they themselves have a name for their weapons. It's just that those in universe can't understand it and so the name is never passed on to the reader.
The Other said something in a language that Will did not know; his voice was like the cracking of ice on a winter lake, and the words were mocking.
A Game of Thrones, Prologue
Lastly, it is worth noting that the Others are deliberately shrouded in mystery and so small details like that are not going to be revealed especially when larger details haven't been.