The right term here would be "patience."
The noun "patient" (plural: "patients") means someone who is undergoing medical treatment. However, the adjective "patient" describes someone who can wait calmly. And, perhaps even more confusingly, the noun "patience" is the quality of being able to wait calmly.
If you want to thank someone for waiting calmly, you can thank him for his patience. You can thank a medical patient for his patience when he has spent a long time in the waiting room.
This is correct
Use this phrase to indicate that you appreciate that someone has been waiting.
Some examples from the Internet:
This is not correct. Don't use this phrase.
You probably want to use the word "patience" here instead of "patient." The word "patient" can be an adjective or a noun, but as a noun, it means someone who is undergoing medical treatment.
Some examples from the web:
If you are dealing with technical assistance or some other support line, the phrase “thank you for your patience” will often be used to apologize for wait times, even if those were not long. Alternately, if you are seeing this in a letter informing you about something that has yet to happen, the meaning is more like “it may take a while for this to happen, and we apologize in advance.” In short, though, this phrase can be taken to mean that something is taking longer than might be ideal and the person speaking or writing to you about that is apologizing for it.