How can a person with total renal failure, who is on dialysis, pass small amounts of urine?

Residual kidney: Yes it is called the residual kidney function, meaning that kidneys ahem still making producing some urine. On hemodialysis this function is lost rapidly however on peritoneal dialysis one may maintain that function for a long time.

Residual function: Patients can be dialysis dependent, meaning that they have end stage renal disease, but still have a few functioning nephrons (functional unit of the kidneys) to produce urine. They just don't have enough to filter out toxins, excess volume, and maintain acid base status.

Here are some ...: "Total" kidney failure means the impaired existing kidney function does not suffice to maintain homeostasis of fluid, electrolytes, acid-alkaline balance, and other functions such as blood production, blood pressure, etc. These adverse effects are not all in the same proportional pace, so resulting in various urine excretion despite requiring dialysis. The urine from such patients is diluted...