Farewell, thou child of my right hand, and joy (1)
The speaker calls his son the "child of my right hand." "Right hand" implies all sorts of things. The "right hand" is the "good hand" (the left is the bad or "sinister" hand). It's almost like saying "this is my right-hand man" (and, in the Christian religion, various prayers say that Jesus is seated at the "right-hand of the Father"). The speaker clearly holds his son in high regard.
My sin was too much hope of thee, loved boy (2)
The words "loved boy" sound a bit cold. Yeah, sure the word "loved" is in there, but still. Oftentimes, poets will lengthen a word like "loved" so that it has two syllables (they do this for metrical purposes) and is pronounced luh-ved. Here, however, it is only one syllable, as if the speaker doesn't wish to emphasize his love.
"[…] Here doth lieBen Jonson his best piece of poetry" (9-10)
The speaker considers his son his "best piece of poetry," his best work of art. Clearly, he loves this child a little more than the others, or thinks he's a little more perfect than his other children (if he has any). He could be just saying this, or he might have more love for his first-born. Nice.