With most legal matters, there are two kinds of fees: the fees you pay for your attorney’s time, and the fees you pay the government for filings, registrations, etc. When it comes to copyright registration, the government fees range from $45 to $500 depending on what you’re registering. You can find more information about fees on the US Copyright Office website.
The attorney’s fees for copyright registration will usually be based on the amount of time the attorney has to spend registering the copyright. While many will charge flat fees, the price is likely based on an hourly rate for an anticipated amount of time the attorney will spend on the matter. The amount of time may vary depending on what the attorney has to do to register your copyright: determining what kind of registration you need, getting specimens, preparing the application, obtaining answers from you for certain questions, and so forth.
All that said, I would expect the majority of copyright attorneys to have an hourly rate somewhere in the range of $150-$300. If you’re in a major metropolitan area (LA, NYC, etc.), it’ll be closer to $500 per hour. Smart and efficient attorneys will offload some of the work to their paralegal, and paralegal time costs anywhere from $50-$150 per hour, again depending on the region.
For the most basic copyright registration (one author, one work, not a work for hire), a reasonable flat fee would be on the order of $200-$400, including all attorney and government fees.