I had the good fortune to spend 3-1/2 days there about a year ago. I
love the city.
What to see depends on their interests. Unfortunately, with only one
day, there are really two cities to see, Buda and Pest, split by the Danube.
Are they interested in photography?
If they get a really early start and use taxis, they might in a day be
able to see the following, which are all central places in Hungarian
history:
Buda (west bank): Historic Mathias Church and royal palace (they would
only have time for the palace grounds and exterior). Also they can walk
the medieval streets in this area, called the Castle District.
Pest: St. Stephen's Cathedral has one of the most magnificent interiors
I have seen anywhere. Here's a pic:
http://pug.komkon.org/05may/stefndom.html
They can ascend to the front top of the cathedral and have a wonderful
view. One can also just walk the streets in this area and enjoy the
architecture. It is a short walk to the Danube
In lieu of one of the above, they could walk across the historic Chain
Bridge and get a wonderful view of the Danube and the magnificent gothic
Parliament building.
It's too much for one day, but one or two of the above are what I would
recommend.
Joe