I've also safely grown LVM volumes online, but /var is a bit more trickey since a lot of stuff runs straight from it .So I don't want to attempt this if it's going to break the whole system.
OK.. get you now..
The main thing to keep in mind is that you'll need a little bit of free space in order to grow /var. I grow /var often on many systems. There's also a possibility that you'll need to run tune2fs to grow the journal to allow a larger maximum filesystem size. This cannot be done with /var online though.