Is sound the best way to transmit a signal under water?
Yes, it is called acoustic communications. Iterative Carrier Frequency Offset and Channel Estimation for Underwater Acoustic OFDM Systems is an example of a paper that uses orthogonal frequency division multiplexing (OFDM) in an underwater acoustic channel.
EDIT:
Note that you wouldn't call it a SONAR any more because SONAR stands for SOund Navigation And Ranging, whereas this is a communication system, just like you wouldn't call your cell phone a radar.
EDIT 2:
Yes, the signal could be converted to a radio signal at a repeater. The repeater would ideally be floating in the surface and have a transducer (which is basically a microphone or array of microphones) on the underwater side to receive the sound waves. The transducer converts the sound waves into an electrical signal, which can be filtered and/or amplified, and then sent to an antenna for radio transmission. The antenna would likely need to be above water because radio waves don't propagate very far through water.