The three most prominent arguments in favour of the Royal Family are: 1) They are great for tourism; 2) They are good for diplomacy; and 3) The institution provides a check on government.
On (3), I'd give you. Any government needs a check, a level of governance above it, in order to limit the risk of tyranny. Granted, the role of our Royal Family in this is almost symbolic - the chances of the Queen rejecting a Prime Minister's request to form a government after an election are slim - but the fact that constitutionally she can do that is enough to provide that check and balance.
On (2), I'd definitely give you that for the Queen, Princes Harry and William, and Princess Anne. They are probably the best diplomats we have in this country. But the others - Charles, Edward, Philip, Andrew, and the hangers on like Andrew's kids' - are not. In fact, they've probably been more damaging than helpful when employed on diplomatic missions.
But point (1) is, to be frank, bollocks. Even Visit Britain - the UK's official tourism body - has said that Royal Tourism doesn't account for much here. And you just have to look at France as as a great example of why the notion is complete BS; the Palace of Versailles get's 17 millions visitors compared to Buckingham Palaces 13 million per year...and they got rid of their royal family over 200 years ago...and their palace isn't even in a major city, let alone a capital one.
If we got rid of the monarchy, tourists would still come here. In fact, we could then open up all the palaces fully to tourists, which would probably attract even more.