I want to build a room in my attic space.  The joists are 2 x 6 and the house was built in 1949 so they measure 1 5/8 x 5 1/2.  The room will just be a getaway room for my grandkids.  There is a center support wall beneath the attic.  How do I determine how far out from the center wall I can go on each side and still have sufficient strength with the 2 x 6 joists for a 30 psf live load?  The attic already has a floor and I don't want to have to rip it up to put in soldier joists if I can avoid.

you do not have a very high load rating for a single 2 x 6 joist, my calculation works out to the following for a 2 x 6 with a 30 psf live load with a 12" between centres you can go 8'9", with a 16" on centre you are down to 7'6", and with a 24" on centre you are down to 6'0" makes for a very small room, you would be better off sistering in another 2 x 6 on all joist that would be under your floor, if you insist on stay with just 1 2x 6 be sure to span as many joist as possible, and be very very careful of how much weight you put up there, or it may end up downstairs

You have joists that are 2X6's...and they're 16 inch on center?

If the lumber is pine, and the flooring is 1/2" or better tongue and groove flooring, you can have a safely expect to exceed over 100 lbs per square foot (according to the 2006 building code, table 2308.10.3(6).

BUT - I need more info. Such as the species (if known), span (O/C"), load bearing wall(s) beneath, etc.

Yes the joists are 2 x 6 that are 16" oc.  I beleive they are fir rather than pine.  There is a wall down the center of the house that is a load bearing wall.  The house is a ranch that is basically 30 x 60 and has brick outside walls.  The joists sit on the outside wall and the center wall where they overlap.  They do not go all the way across the 30 ft.  The floor is 1 x 8 planks that are mostly 12 ft long.  They are not tongue and groove.  They are nailed into every joist and cover the entire attic.

30 PSF live load for 2X6, 16" OC

Maximum floor joist spans for a dead load of 10 or 20 PSF are:

Douglas Fir-Larch    11' 4"
Hem-Fir                  10' 9"
Southern Pine         11' 2"
Spruce-Pine-Fire      10' 6"

Dead load - building material.
Live load - furnishings.

This is from a table in the building code (section 2308).