Welcome Tom!
From our own world to Westeros, it seems that naming a child after an important person is a way to honor that person and to try and shape the life of the newborn. Remember Cersei was horrified when the Stokeworths suggested naming Lollys' child after Tywin; naming the offspring of an imbecile and her peasant raper after the greatest lord of the realm would have justifiably offended the Lannisters. And Cersei was later incensed when Bronn named that child Tyrion; she took it as a slap in the face, because that's how it was meant.
So when Robb was named after Ned's best friend and brother in arms, I'm sure Robert was pleased. But how was Jon Arryn, the great lord of the ancient Andal family supposed to react when Ned named his ******* son after him? I think the Arryns (and the rest of the world) might have expected Ned to name a subsequent legitimate son after Jon Arryn. Was Jon Arryn insulted by this naming?... For Pete's sake, he just rebelled against his king and fought a war for Ned, and Ned repays him by naming a ******* after him...
Three thoughts come to mind as to why Jon Arryn might not have been angry. First, Jon knew Ned's character. He raised Ned and Robb as the sons he never had. He loved them, but he was not blind to their natures. He knew Ned was always prickly about his honor and about letting others have theirs as well. He knew Ned never indulged himself in passive aggressive behavior... if Ned had a problem, he dealt with it himself. So naming the child Jon was not intended as an insult.
Second, Jon knew the value of family to Ned. Ned seemed very close to his father (I assume his mother also, but she's never mentioned), Ned looked up to Brandon as his hero, Ned was extremely close to Lyanna, and Ned loved and protected Ben. Now for Ned to keep his ******* and to give it a special name... well, this meant that Ned was going beyond his legal and honorable obligations, ie. the child was extra special to him. Jon Arryn must have trusted that Ned had a good reason for naming the child after him and not naming it after the immediate family.
Third, Tom O'Sevens suggested this reason in the last post... Ned told Jon. When Robert won the rebellion and was crowned, Jon immediately became his Hand. Jon Arryn was in a position (if Jon Snow was Rhaegar's son and if Jon Arryn knew about it) to divert Robert from Jon Snow's parentage. I'd say after Lyanna's death there were only two people that Ned trusted implicitly... Jon Arryn and Howland Reed... he may have later come to trust Benjen completely, but that's another story.
Jon Arryn may have known the truth about Jon Snow, but then he might not have. Ned never told Catelyn.