What does 帮忙 really mean? 妻子要丈夫帮她洗碗，丈夫不好意思拒绝，就把儿子叫到面前说：“孩子，现在让你练习洗碗，以后可以帮太太的忙。“ 儿子不以为然的说：“不必，以后我可以叫我儿子洗。“ 哈哈！

I found the expression '帮太太的忙‘ unusual. I presume it means here either a)  'help your wife when she is busy' or just b) 'help your wife'. 

Do 帮 and 忙 work together here or are they 2 separate words? 

帮忙： help， lend a hand

Does 帮忙 really mean: 'help (someone) when (they are) busy'??
Your correct, 帮忙 just means to help (someone). It can also be split apart two word, 幫 as help and 忙 as busy.  

The way the sentence is composed is in a more creative way, that uses the meaning of the characters creatively, both combined and uncombined. However, it's a common technique that shouldn't be over thought.  

你可以幫一個忙嗎? vs 你可以幫忙嗎? are just two different ways of asking if you can help.