Mongoose is great for cheap but reliable. If you feel like spending a little more money than you would with a Mongoose, go for Trek. I used to have a 20" BMX bike that was a Trek, and it had never had any problems besides bent rims and popped tires (normal for that kind of use.)
I was a bike mechanic for 4 years actually. Probably the best bike for little coin are what have already been said. Also look into Cannondale's, all of there bikes are handwelded in PA. They start about $500 I believe (or used to) for one of there lowerend models. Also comes with a lifetime frame warrenty. Like said above, look for shimano or SRAM components. Also look for sealed cartridge bearings in the bottom bracket, hubs, and head set. If you have any questions you can hit me up on aim.