1) Tradition and constancy. A monarchy represents something that has been around for 1,000 years. A republic can never have that, as any head-of-state it has will be relatively new.
2) The monarchy doesn’t cost very much.
3) The Queen can never be corrupted, because she is too rich to be bribed or bullied.
4) The monarchy is the basis of the Commonwealth. There are 83 countries currently in the British Commonwealth, and the Queen is the head of the Commonwealth. Every four years, all 83 countries send representatives to a meeting to have an intellectual exchange. The Queen, of course, is a very experienced politician, having been in power for about 60 years. It is very useful to have a forum for international discussion, and so the Commonwealth is a positive thing.
5) Monarchy is one of the only uniting factors in the United Kingdom. Few things actually unite Ireland, Scotland, England, and Wales, and the monarchy is one of them. Unity is a good thing.