너 팔베개나 해주게 = to make you an arm pillow (ti.e. o put my arm under your head like a pillow) '-ㄹ게'로 끝나는 완전한 문장이 아니라 '동사 + -게'로 끝나는 부사구라서 그래요. It is just an adverbial phrase indicating one's intention (-게 means intention or purpose, similar to -도록). There must be a sentence or action leading to this phrase for it to make sense. For example, you can imagine a situation like this. A: 왜 내게 가까이 와? (Why are you coming close to me?) B: 너 팔베게나 해 주게. (Just to make you an arm pillow) = 네 팔베게나 해 주게/주려고 (가까이 온 거야). To make a normal sentence of "I will make you an arm pillow", it should be 줄게, as you said. 나 in 팔베게나 is a particle used for either dismissing something as unimportant (할 거 없는데 점심이나 먹자) or noting something for its magnitude(체중을 5키로나 뺐다) - these opposite uses are distinguished with a different tone. In your example, it's used in the first sense, like "Nothing special, I just want to ...".