I understand that Lyanna Stark was abducted by Rhaegar leading to the arrest of Brandon Stark. I don't really understand how this relates to Jon Arryn's wards though.
The Mad King killed traitors' families. When House Darklyn rebelled, he extinguished them and house Hollard, (Except of course for Dontos).
He demanded the deaths of Robert and Ned after he had killed Brandon and Rickard. The Mad King suffered from paranoia and executed Rickard and Brandon for treason without a fair trial. When Rickard demanded a trial by combat, Aerys named fire as his champion.
Ned was Lyanna's brother and Robert was her betrothed, both loved her and the Mad King knew that they would start a war to get her back. Aerys wanted Jon Arryn to kill them before they could call their banners. If Jon Arryn had done this, it would have prevented the rebellion.
Lyanna Stark was abducted by Rhaegar leading to the arrest of Brandon Stark
Brandon Stark was arrested because when he learned that his sister was abducted, he went outside the Red Keep and asked for Rhaegar to come out and die. The latter wasn't there, but the Mad King was and had him arrested. This led to the death of both Brandon and his father Rickard.
Keep in mind that before Lyannna was abducted by Rhaegar, she was engaged to Robert Baratheon. So, even after Brandon and Rickard were killed, it was only expected for Robert to want revenge and fight to take his betrothed back. So, I guess that you can say that he was being thorough with the threats against his seat on the Iron Throne.