Stayed in Budapest two weeks ago and stayed in a lovely self catering apartment. Classical old building but renovated really well, modern, spacious and truly of a 4* hotel equivalent yet €60 per night.
Exceptional value for money and unique accommodation to boot. Worth considering.
Andrassay Utca apartment. Check out www.budarentals.com
Updates
I travelled to Budapest last weekend with some friends and stayed in the Parliament apartment I found on budarentals.com. This apartment was really well situated, had plenty of room for four of us and was great value for money. Highly recommended! I stayed at the Parliament apartment too, and you can't get much more central than that, close to all the attractions, it was really worth the cash.
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www.budarentals.com/Budapest_Apartments_for_rent/Basilica_District_V.aspx