I know that "online" means a person is reachable over the Internet.
Can I say "staying online" while speaking about phone calls?
Example:
Thanks for staying online. (Thanks for not hanging up with me).
We say:
I'm going to go offline this weekend.
This means that we are going to put away those devices that keep us online.
Most commonly these days, it refers to our mobile phones. And offline does refer to both internet access and phone calls.
We also say that we're going online, meaning that we are checking Facebook, Twitter, messages, and so on, and generally making ourselves available to others—including by phone.
So, staying online can include phone calls, but it includes a bunch of other things too. So, if only talking about a phone call, I wouldn't use it in that sense.
I would instead say one of the following:
Thank you for staying on the line.
Thank you for staying on the phone.
Thank you for holding.
Thank you for waiting.