the thermocouple is made from two wires that produce a voltage at their junction depending on the temperature. Actually, all conductors produce a voltage when under a thermal gradient; the thermocouple uses two different metals to ensure that there is a DIFFERENCE in voltage between one side and the other - the difference can be as little as 1 microvolt per degree (up to about 70 microvolts per degree)
In order for them to operate correctly, the two legs of the thermocouple need to be the same length and similar metals. For this reason, I would avoid "just twist[ing] and crimp[ing] a new thermocouple junction" as this will result in new metals and conduction paths in the hot part of the thermocouple. As you can see at the diagram
http://en.wikipedia.org/wiki/Thermocouple on the right side of the wiki article on thermocouples - you want the join to your measurement equipment to be at a cold junction, not in the hot zone, for best accuracy.
If you feel like trying it, you can certainly do so faster than you could order a new one. However, I would want to be very sure the temperatures its measuring are accurate when done.