The main reason for using Constantan (made from copper and nickel) is its temperature coefficient. Look at the following graph (it is in german, sorry) - taken from . It shows specific resistivity and TempCo over nickel percentage.
Constantan has fairly high resistivity, which is good because you do not need so many squares to get to reasonable resistances like 120 Ohm. It also has a very low TempCo which you can even fine-tune during production to adapt to the thermal expansion of different material.
I did not look at Wikipedia, but if it says one wants high gauge factor metals for good strain gages, it is wrong. In fact you want a gage factor of approximately 2 because this means, that geometry effects dominate over electron mobility effects as the physical reason of the gage factor . Electron mobility is heavily temperature dependent, therefore you don't want it. So a gage factor of two means low temperature dependancy, this holds for Constantan and NiCr (Karma and variants).
Beware: This is only true for metals. Semiconductors are a totally different story.
 G. Arlt, The sensitivity of strain gauges J. Appl. Phys., vol. 49, no. 7, p. 4273, 1978.