Over time, graphic designers will want to, or have to, raise their hourly rate for design work. When taking on new clients, this is easy, because the client does not know your previous rates. When raising rates on existing clients, this can get a little trickier. You don't want to lose a client over a $15 an hour increase, but you also often cannot continue to keep a client on at an old rate.
When raising your rates on an exisiting client, "honesty is the best policy" certainly holds true. Explain to the client that you have kept their rate as-is as long as possible, but due to increased costs of operating your business you have to raise their rate. If you have a range in your rate, you can offer them the lower end (even if an increase) and let them know you are doing this. A good way to "ease" the client into the new rate is to offer them one last project or bill at their old rate, or give them a discount off the first bill on the new rate. Of course, these options are OK if you can afford them, but they are certainly not required. A client that respects you and your work should not be surprised at an occasional raise in costs.
If a client refuses to pay more, it is up to the designer if they wish to continue the work. If you enjoy the work a lot, if the client is easy to deal with, or if they provide steady income, you may wish to keep their rate and approach the topic again at a later date. If the client has caused problems for you in the past, and you have plenty of other income, you might want to consider moving on. This is all of course a personal decision based on your relationship with the client and how you may be affected by the loss of income.