When you say flame is blue, I presume you mean the pilot light while you are holding down the start button - which means you are getting gas OK to the gas controller and the pilot.I presume the flame is steady and not wavering all around, which fails to keep the thermocouple hot enough to work. The flame should be about 3/4-1 inch tall in most furnaces - if a tiny flame, you might have to adjust the pilot gas flow a bit higher - google for the owners manual for your model at the manufacturers website or a site like Sears.
I presume you know it can take up to a minute or so for the thermocouple to heat enough to keep the flame going when you release the start button ? So hold the button down for at least a minute after lighting the pilot before releasing the start button.
Is the thermocouple tip sticking 1/3-1/2 its length into the blue of the flame ? If only the very tip is in, or only into the very pale almost transparent green-blue-red base of the flame, then the air flowing by to the flame can keep it cold enough it may not get hot enough to tell the gas controller that the pilot is lit.
IF not that, unscrew the flex tube from the gas controller. Then slide the fastening nut back a bit on the tubing, insert the tube tip up into the hole, MAKING SURE TUBE IS STRAIGHT AND IT IS GOING ALL THE WAY TO BOTTOM and not just hanging up on the threads. Then, while holding the tube in tightly, slide the nut up the tubing and screw it in, tightening down snuggly but without overtightening - just barely wrench tight - do not reef on it, just a bit more than hand tight. If you overtighten you can deform it, causing it to short out. Try to light pilot again. Frequently, when people put the tube and nut in together, the "Button" on the tube end hangs up on the threads so it does not touch bottom, so the rounded button tip is not touching bottom in the controller, so no signal gets through.
IF not that, check for any kinks in the tubing - if kinked, it is probably toast. Occasionally straightening it out will work, but usually not. Curves are fine - kinks not.
I have seen one brand of electronic thermocouple where, for some unfathomable reason, the outside of the tubing is "live" and the part where it screws into the pilot fixture is insulated - so if it touches any metal it does not work, so make tubing does not touch any metal between pilot fixture and gas controller.
IF no go on these, it is possible the gas controller is shot - but I would try a brand new thermocouple first to be sure - at worst you will be out $5-10 or so rather than hundreds for a new gas controller installed, and will have a spare thermocouple for next year if it does turn out to be the controller, so you will really be out nothing in the long run and may save several hundred dollars.
I would give about 50:1 odds it is a thermocouple positioning or tightening or kinking problem - gas controllers/valves are really reliable, and usually go out by developing small leaks in the seals around the knob shafts, not by internal workings failure. Could also be a defective one - I have run into two defective Honeywell Universal thermocouples in the past year - flat out did not work right out of the package, but a new replacement one did.
IF all that fails, then an HVAC contractor (or a plumber if a hydronic or steam heating boiler). Note - under law in most areas, they will NOT attempt to repair a gas controller - they will only replace it.