Popular attraction in birmingham

The islamic art is meant to be worth seeing, if they will allow you into the city...

More seriously depending on the childrens ages..Cadburys World.
Cadburys equals chocolate ..

Cadbury World - no longer British but now owned by that so very small artisanal company and oh so American, Kraft - is easily reached by train...

Are you staying in Birmingham city centre? Sorry, I have never used taxis as I always drive there, but if you google "Birmingham airport taxis" it shows the email addresses of a number of companies. It will be cheaper to book in advance, as most companies will have flat rate fares from the city centre to the airport. There is a taxi rank at the airport, but these will be more expensive than pre-booking. Can your hotel recommend a cab company that they use?

The Birmingham International rail station is at the airport and provides c 10 trains an hour into the city centre and unless you are arriving at peak train times (before 9.30 am), this will be a cheaper option than getting a taxi, unless you then have miles to travel to your hotel.

Cadbury World that has already been suggested is c 5 miles outside the city centre and can be reached by train from B'ham New Street to Bourneville then a short walk. The train takes c 25 minutes.

There is plenty to do in the city centre itself, depending on the ages of your children. The Jewellery Quarter is located immediately north of the city centre and in addition to housing many jewellers is also home to the Pen Museum (free entry) which has children's activities, the Jewellery Quarter Museum and the Coffin Works (not as gruesome as it sounds - entry charge). This area can be reached in just over 15 minutes on foot or via a bus from the city centre.

If your children are older, they may like the National Motorcycle Museum which is out near the airport.