How could a male noble make a name for himself if he’s a cousin of the one holding the lordly title and he doesn’t marry an heiress? Like, he builds up a reputation and respect, but I don’t know how.

Well, one could go the military route – work very hard to become recognized as a talented knight, win glory and prizes on the battlefield or the joust, maybe become a professional man-at-arms.

Or one could go the political route – go to court, become known as a courtier, suck up and flatter one’s social superiors, try to get a government office, etc.

Or one could join the church and try to get oneself a goodly benefice.