I guess it also depends on the date (winter or spring?) and your taste, budget, age... I guess none of you is a local?
As the above suggestions indicate, the hilly nature of Buda offers exceedingly great views to the Danube and its jewels, if weather and inner fire permits, some locations are a bit touristy though which, for me, is not so compatible with 'romantic'.
E.g. I think the restaurant at 'Vajdahunyad Castle' is not only expensive, full of tourists, but I heard it is also bad - you can still have a tea - the view is unrivalled.
The 'Sikló' train descending from from the Buda castle is another special experience; from the lower end you can walk across the Chain Bridge e.g. into the truly beautiful (and pricey) hotel-café in Gresham Palais)
http://en.wikipedia.org/wiki/Budapest_C … _Funicularhttp://en.wikipedia.org/wiki/Gresham_Palace
Or, starting from a different bridge, walk from Hotel Gellért to the Cave chapel,
http://en.wikipedia.org/wiki/Gell%C3%A9rt_Hill_Cave
and further up (I think 15 min to the top is reasonable).
MY personal *must* is an evening/night boot trip - I know it sounds too touristy, but it is a tried and tested ;)the view is even more breath-taking. That can be a bit more expensive though if using tourist boats.
However, the public transport company (BKV) is also operating Boats, afaik even during winter (I havent tried them yet):
http://bkv.hu/en/boat
Simply walking along the Pest promenade (near Börösmarty tér) gazing at the Buda side, or, if cold, taking the Tram 2, is also an option. Restaurants along that mile will probably all aim at tourists, but the great view is there, too.
Lets not forget about the date-paradise (of a certain age-group) that is the Mũjég ie. the ice skating ring in Városliget with the amazing building and view to Vajdahunyad castle (just be aware that the median age is below 20...)
http://www.mujegpalya.hu/?lang_id=2
...and - how come we mention this only now? - walking on *Margaret Island*, THE romance spot on summer evenings... granted, it has less view, so you may get bored;)
Nah, just kidding, just sit/stand at teh south tip, below the bridge, and look at the Parliament. Btw, that view was a big factor I decided to move home after 20 yrs.
Then walk on the newly renovated Margit Híd to Pest, and
to regain lost calories, there is nice new café 'Maródi Cukrászda' at Jászai Mari tér at the end of Margit Bridge. (You also may like to walk further into Újlipótváros i.e. the part opposite to Margaret island)
Or, 5 min. from the Margaret Bridge, find a small but cute cinema-café called 'Kino' hidden at the right side of Vígszinház (the young owner worked in the UK for years). The cinema (which I never tried) is off-mainstream, showing artistic movies too
http://www.akino.hu/kavezohttp://www.facebook.com/pages/Kino-Mozi … 4395613934
A totally different thing, perhaps not traditional romantic, but worth mentioning, is Corvintető, a nice underground bar/danceklub on the roof - google some pictures! I don't know if the roof area is open in the winter though.
None of the above is particularly expensive... if you ignore Gresham Palace, but walking in is for free.
There are a lot of 'floating restaurants' in the boats on the Pest side with view to Buda Castle (don't know about buda), of course not exactly 'budget' for most of us, but the view...
I just mention in passing as it is obvious that, Budapest being the ex-second-capital of the Habsburg Monarchy, there are plenty of classical, turn-of-the-century cafés and restaurants: Café New York ('must see' its Hotel Boscolo now), Café Gerbaud,... some *are* breathtaking, even before you get the bill, just google some photos.
Finally, I googled a different list:
http://budapestiprogramok.blog.hu/2009/ … dapesten_1
I think that may be enough for now - let us know how it went;-)