Was Robert Arryn named Robert to honour King Robert? Whose choice was it?
"The boy is my namesake, did you know that? Robert Arryn. I am sworn to protect him. How can I do that if his mother steals him away?"
In answer to the second part of your question, there seems to be no canon description of who named him but there's also no reason to assume that it wasn't his parents; Jon Arryn and Lysa Tully.