Cheap s not a good Idea
The main problem I have with a cheap bike is that you are taking a bike that does not have much of a shelf life and adding a lot more power and vibration. If you start with a decent bike and add a engine to it you chances of lasting more than a few miles is much better.
The bike I am currently using is a modified Giant Mountain bike. I have also had to double bolt and lock tight just about every fastener on the thing. I think that it would be a good idea to start with a mountain bike frame(because they are overbuilt) and add parts accordingly. I just have a eight speed free hub that I had to rebuild after a year because of the stress. I also just went with a single front chainring because if the motor is working I don't need the lower gearing.
I used to run a bicycle shop a wile back and have found that department store bike have a lifespan of about four months of use without proper adjustment. Its also much harder to get a cheap bike to the point were they are ridable. I just spent about four hours on a friends cheap mountain bike just to get working right. A lot of shops wont even work on a huffy even if you decide to bring it in.
If you do decide to get a department store bike I would just get it in the box and take it to a bike shop and have them assemble it. That way at least it started out in good shape.
mike
motorbikes.com
