Some missions are coded so that you can choose weapons before starting the mission. The mission has to have been designed to allow this and to specify what weapons choices are possible.
To see if you're playing such a mission, before you start playing a particular mission, click on the GEAR tab in the mission notebook. You'll see pictures of your primary and secondary weapons, etc.
Click on a weapon's pic and you'll get a list of alternatives you can select, if the mission designer chose to allow you to do so.
As an example, start the Steal The Car SP mission and follow the instructions above. The mission allows you to change the default M16 to either an XMS rifle or an M60 MG.