It would be really helpful if someone suggests an Op Amp which takes an input voltage in mV and the output of +-5v. If there's an evaluation board that comes with the Op amp which takes the mv input and gives a +- 5V output, then it would be even more helpful.
Assuming you have a supply of more than ±5V, you should be able to pick pretty much any op-amp and it will be able to do what you are wanting. It is all dependant on the feedback resistors used.
It will likely be much cheaper to stick a circuit on breadboard or a bit of stripboard and use a DIP socket to change the op-amp than it would be to buy a EV board. You could have female sockets for resistor pins to plug into as well so you can change the resistor values in the feedback loop.
EV boards can be upwards of $20 (some better op amps will be much more) whereas you can probably build your own with interchangeable parts for less than $5, and you'll be able to see virtually any op-amp can give you your desired output with a mV input.