Excuse the graphics skills but attached now is a drawing:

I'm going to speak generally about this because I'm not sitting in front of my statics or strength of materials books and I not intimately familiar with what you're doing.  I'm also not a design engineer, but I am a welding engineer and I'm hoping to give you enough info to get you going on your project.  No warranties expressed or implied, see stores for details. 

What you're thinking through is the effect of the cross section, formally called the section modulus, on the behavior of the beam. When you take a square tube and cut it like you're describing, you create a beam with two cross sections.  The first cross section is of the original beam and it exists in all the areas where there isn't a cut.  The other cross section is the remnant materiel at the location of the cut.  In your case, that's a channel shape (assuming I'm thinking of this right).  So at first glance, you want the channel to have sufficient cross section to match the section modulus of the original square tube.  You can do this by making the channel deeper (high side walls), making the bottom thicker, or making the sides thicker (think like gussets - this is the way I'd go because it's straightforward and cost effective)

But, that misses something that _might_ be important (I doubt it in your application, but I'll include if for completeness).  When the tube transitions from a square to channel cross section, the neutral axis of the cross section has to "jog" around the cut.  This causes the bottom of the cut to be a stress concentrator: (Figure 2 here, your saw cut is like the notch on the far right: http://www.teachengineering.org/view_activity.php?url=collection/cub_/activities/cub_tower/cub_tower_activity1.xml).  If you have alot of load, any cracking will initiate at this location because it's the proverbial weakest link.  You can do some things to combat this.  First, drill a hole where you plan to stop your cut.  This will change the square end of the cut into a round which helps with stress flow (look at how stress flows nicely around the circle in that link above).  The larger the hole, the lower the stress concentration: (third picture down: http://en.wikibooks.org/wiki/Strength_of_Materials/Unsorted_topics).  Second, with the gussets I mentioned above, there's more cross section at the transition point of the cross sections, so the stress gets divided up between the box section with a gusset and the channel section with a gusset.  Think of it like adding two lanes to a highway at the location of a really busy interchange. 

To figure out the thickness of the gusset, you'll need to look up the section modulus formulas for a box tube and a channel (NOTE: because a channel is not symmetrical, you have to be careful about what section modulus formula you use - they will be different for loading that bends the sides vs loading that bends the bottom of the channel).  If you do an online search for section modulus you'll find the formals, they're pretty easy "plug and chug", nothing too exiting.  Input your numbers and see how much lower the channel section modulus is.  Keep units consistent!  If you start with inches, you gotta do everything in inches.  Then, start increasing the thickness of the sides till the section modulus matches or exceeds the modulus of the box tube (this assumes you use material with similar mechanical properties for both the tube and the gusset, you can play games with material strength, but then things get a whole lot more complicated).