The ring probably symbolizes a simpler time for Frank. As we saw in his class reunion episode (Season 1 Episode 8), Frank was quite different and more idealistic back in school. He sang in a barbershop quartet, was in a loving relationship, and basically ran wild and rambunctious with his schoolmates. read more
It wasn't until the very end of the episode, when Frank has left his school and friends behind, did he start talking to the audience again. At the civil war site, he discovered that an ancestor of his was buried in the memorial. read more
To me, Frank was only tapping his ring, not his knuckles directly. This he was only symbolically following this father. He wasn't fully preparing himself for the fight. By burying his ring he will have to hit his hands directly and toughen them up for the big fight with Tusk over who will control the president and the future. read more