If Thomas Paine (T-Paine) felt so strongly about his ideas and thoughts, why didn't he put his name on the cover of Common Sense? Was it cowardly for him not to do so?

It was most likely that he didn’t want his name attributed to Common Sense because of possible repercussions and punishments (which could include prison or even execution.) Advertising his name all over the pamphlet would be the equivalent of taping a wanted poster of himself to his chest and prancing through a street full of loyalists and British soldiers. Any logical person would try to avoid that as much as possible.

Was it cowardly? I don’t think so. He was only trying to protect himself from incrimination while also making his thoughts and beliefs clear to his readers. If the British caught him, copies of the pamphlet would have likely been discontinued or burned, so his message wouldn’t have gotten out to many people. He wanted both himself and his ideas to survive.