It turns out that defining predefined field variables in CAE is a feature that was added in the 2018 version. The steps that one needs to follow when creating a field-dependent material property are the following:
1) In the Property module, make your Material dependent on a Field Variable(s) by specifying the number of field variables. In the table you will have to map the material property to the field variable.
2) Create the Field Variable: In the Load module, go to "Predefined fields"->Create->Other->Field and specify the desired distribution, which can be one of several options, including an Analytical Field. This analytical field in my case is the spatial distribution of the expansion coefficient - a function of X,Y,Z.
In case you have an earlier version of Abaqus/CAE, apparently you could use a temperature definition for the Field, and modify your input file to reroute to the field variable. I haven't tried this approach, but it should work just as well.
A couple people suggested parts of this and then I finally figured the rest out through some trial and error. Thanks!