What should I use to frame walls inside a concrete structure?
There aren't many options for framing cheaper than basic 2x4 lumber.  You could go down to 2x3 which is sometimes cheaper but you'll find that interior doors and insulation are generally sized for 2x4 walls with 1/2" drywall. 

You could also consider metal studs.  These are available down to 1.5 inches in thickness (fine if all you want to do is hide the concrete.)  The cost is often slightly higher than wood, but you won't get mold if the space is still a little damp.  They are also usually available in longer lengths (owing to their primary use in commercial spaces) and they are always dimensionally straight.  Installing metal studs goes really fast once you get a hang of it, but again doors can be difficult.