My pilot light has been relit twice. According to the person doing the job, he confirms that after ten minutes, the pilot has remained on. However it seems to have go off after this period. What does this mean and what can a person expect to pay in costs for parts and labor? Thank you!

How old is your tank? Could be a vent probem.

Thermocouple can be cleaned with sandpaper or replaced. The copper thing that sits in the pilot flame.

It is unknown as to what the issue could be, while very likely a thermocouple it can be anything including gas supply.