I went with my family a couple of years ago, and we stayed in a boat hotel on the Danube ... not bad at all, and very reasonable (especially when we offered to pay in cash -- got a nice discount).
Here's the place:
www.budapesthotelstart.com/budapest-hotels/fortuna-boat-hotel/hotel-summary.en.htmlIt's near Parliament, with a view of Margaret Island. I think we paid around 50 Euros a night for a double (in early Nov.), but I could be off by a few.
As others have said, make sure to hit one of the baths -- it was the coolest thing we did. (We went to the Szechenyi Baths,
www.szechenyibath.com, which were less touristy than the more famous Gellert Baths on the river.
Also, hit the huge Central Market Hall, and check out the view from Castle Hill. Also good are St. Stephen's Basilica, the Chain Bridge and the House of Terror (which is rather sobering, but fascinating).