Accommodations in Budapest

Consider: K&K Opera. Excellent location and a fine hotel.

It would be better to define inexpensive lodging. Give a range.
I too recommend K&K Hotel-very nice modern hotel quite central for walking with excellent grand buffet breakfast. But I do not consider it inexpensive at approx $180 per night for 2 persons w/ breakfast. I stayed 4 nights there in 2012.

First, location. I think to get the absolute most out of a visit you have to be well located. Generally speaking this is where I think most people will get the most out of their visit: Get a map, find Andrassy ut (the street that the French used as a model for the Champs-Élysées – or possibly the other way around....??). On Andrassy ut there is a major intersection and metro stop called the Oktogon. Now draw a line from the Oktogon down the center of Andrassy ut and further until you hit the Danube river. Now find a place a block or two above the line or a two or three blocks below the line. I think this is the sweet spot. I have stayed a few times at the K&K and it's my favorite hotel for cost and location; in the center of this zone. But there are others too. Go to Google Earth or Google Maps and hunt down hotels by location and check their websites. Pick one that suites your personality and budget. This town has a lot to choose from, but try and stay in the zone I outlined because there are a lot of beautiful hotels that will kill you with transit time back and forth. Wouldn't you rather be able to walk out the door in the evening and be where you want to be for dinner, drinks, strolling, bars, theater, etc; rather than even a short 5 minute metro ride? As for the apartment. A two bedroom and two bathroom apartment is a rarity. I know the apartment rental business pretty well and I don't know one that fits that bill in a decent area of town. I know of a couple of situations where there are two apartments adjacent to each other and you will have better luck if you can accept that. Depending on the time of the year the hotel will cost 90 euro and up a night and two apartments about 90 to 150 euro a night for the pair. The advantage of the hotel is the service, the advantage of the apartments is the opportunity to live among the locals and get a deeper insight into the culture.

It would be great if we could keep the hotel rate at $100/ night. The k&k does look great , though

2 years ago, we went with one of Rick's recommendations, your "Hungarian aunt and uncle". The location was not hugely convenient, but was on the metro. We paid 80 E / night for 2 Bd/1 Bath in an old apartment house. Clean, well-suited. Our hosts provided an inexpensive parking place for our car as well. The hostess was Maria Battagliarini. They were helpful in many ways for us.

USD100 a night PER ROOM is possible but you want two rooms and two baths. You might check the Opera Garden Apartment Hotel and a Hostel called Wombat. I know people who have stayed at both and have been satisfied. The location of the Opera Garden is "Good" and the location of the Wombat is "Very Good". Most small two person apartments depending on the time of the year will run you 45 to 75 euro. So the 45 euro end is about USD60 x 2 = USD120. If you move out of the center you might do a little better but you have to trade off the cash for time, convenience and experience. If you do go the Apartment route, which is a good option, find an established apartment management company with a good reputation and not just a guy named Gabor Or John who happens to own a few properties. With a respectable management company they have to make sure you are happy to maintain their business; Gabor may have other ideas.

JS, the breakfast is great at the K&K but they stick you pretty hard for it. I bet that was 30 euro a day for the two of you. There are some simple local places very nearby where so you could get a nice breakfast for half that. The convenience and the size of the buffet make it worth the money if you have the money but on a budget, maybe not. Still my favorite all around hotel in town.

Thanks for the info. It helps a lot.

I made reservations (for 4/2014) through Booking.com for Grand Market Luxury Apartments, owned by Pampas Argentine Steakhouse. $67 per night approx. It's across the street from the fantastic Vásárcsarnok Market and a couple of blocks off the river.
There are hundreds of relatively inexpensive accommodations in Budapest.