On my recent visit to Japan, I landed at Narita airport and departed from Haneda airport. I was told that a Suica card was a very convenient thing to have, so I purchased one on arrival at Narita. At the end of my trip, I wanted to return the card at Haneda to get back the money I had put on it, but I was told this was not possible, with very vague explanations. What gives?
There are several companies that issue Suica cards, and a Suica can only be returned to the company that issued it. In your case, you purchased a Suica from JR East at Narita, and you probably tried to return it to Tokyo Monorail at Haneda; those are different companies, and so that is not possible. The same would have happened in the reverse situation: a Suica purchased from Tokyo Monorail cannot be returned to JR East.
For this reason, I always recommend that visitors purchase a Pasmo card instead of a Suica: it is functionally identical as far as short-term visitors are concerned, except that it does not exhibit this issue (a Pasmo purchased from any participating company can be returned to any other). At Narita (resp., Haneda), a Pasmo can be purchased from and/or returned to Keisei (resp., Keikyu), even if you do not use their trains.
By the way, if you find yourself at Haneda with a JR East-issued Suica that you really need to return, you must go to a JR East station. The nearest one is Kamata (take a Keikyu train to Keikyu-Kamata, and then walk a bit to Kamata), but if you have a valid JR Pass you may prefer to take the Tokyo Monorail to Hamamatsucho, since the monorail is free with the JR Pass.