There's one thing I've seen occasionally in poetry (the only examples that currently spring to mind are from Edgar Allan Poe, but I know I've also seen it elsewhere) where instead of using two different words that rhyme with one another, the last word in two lines is the same and the second-last words in those lines rhyme:
Thou was all that to me, love
For which my soul did pine
A green isle in the sea, love
A fountain and a shrine...
'Wretch!' I cried, 'thy God hath lent thee-
By these angels he hath sent thee-
Respite, respite and nepenthe
From thy memories of Lenore...
Is there a word for this? Or at least some way of describing it better than what I've done here?
(EDIT) Re Chappo's comment: I've tried a few Google queries along the lines of 'poem second-last word rhyme' and looked up a couple analyses of the poems linked above (which were sadly heavy on flowery descriptions of how moving the poems were and low on 'this rhyming device is called such-and-such'). I'm not sure what resources exist for answering questions like this, I'm afraid.