Since budget is your main concern, you might consider the off-airport area. At every station on the Keikyu line out of HND from Tenkuubashi to Keikyu Kamata, there are budget-friendly hotels and hostels. Some are newish. Google maps shows the full selection.

Now some of those neighborhoods are quite lifeless, so I don't think they work for someone who is looking to explore a neighborhood in the afternoon and have a nice dinner. If you want to be in a location with a little life, Kamata itself is a commercial hub / entertainment district with places to eat and drink. Some people will say that it's the armpit of Tokyo. Well sure, there are some less-than-salubrious establishments. The area around JR Kamata station is one of Tokyo's top areas for crime... as are sections of Shibuya, Shinjuku, and Ikebukuro. IMO it has a worse reputation than it deserves.

Following day ... You can be at the airport in minutes.