I have a Syrian friend, who has a Syrian passport, and also a visa for Italy. He is in Jordan right now, and he wants to visit the Caucasus. Would the Italian visa let him enter Georgia also?
Yes, he can.
The Georgian government provides a "Do I need a visa?" wizard. It says that your friend would be able to enter on the basis of Ordinance 256, which gives a list of countries "whose visa and/or residence permit holders may enter Georgia without a visa for an appropriate period and under appropriate conditions" "for 90 calendar days in any 180-day period."
You need to be more specific about what kind of visa does he have issued by Italy? Is it a long term visa or a short term visa? Or is a valid residence permit?
If it's a long term visa or residence permit - then he can enter Georgia for short stay without any problems. He'll have to prove that he has enough funds to support himself during his stay in Georgia.