Hi Cellis.
Around what year did they emigrate ? Did they ever came to Canada or United states ? Otherwise, I am not familiar with any tools for Argentina.

This surname (with one "r" does not seem frequent in Italy). Maybe the initial cognome (surname) was slightly different.

Torrioni is a place in Italy.
Sometimes, immigrants took their native place name.

1-You may try there - there are two ancestors with the name - one in the Orvieto region:
http://poster.beniculturali.it/poster/action.do
You may have to register
(see image)

2-This other website is cool and might be useful (it searches multiple websites - don't be afraid, it will open multiple windows).
There seems to be some Torioni's in the Milano region:
http://www.geneaservice.nl/navigator/italy/

Ellis Island website suggest some variants:

Torrioni 97%
Turioni 97%
Toriani 96%
Toriumi 96%
Turiomi 96%
Torianni 95%
Torriani 95%
Turiani 95%
Teriani 94%
Torione 94%
Toriono 94%
Tosioni 94%
Turianni 94%
Turriani 94%
?ariani 93%
Tarianni 93%
Terianni 93%
Terriani 93%
Torrione 93%
Torriono 93%

MAJ