Just sounds like the A/C blower motor resistor or motor itself is overheating and shutting off. Either the blower motor itself is crapping out or the resistor. Does it blow in any selected speed?
Does the A/C compressor (the compressor is along the front of the motor with the serpentine belt going around it's pulley...) continue to run when the blower stops? Pop the hood and check.
You need a good indy shop to look at it and do some testing.
It can get a bit complicated for the novice.
Do you have the Bentley or TIS manual for your car?
A/C Blower Motor Resistor....