I have a Gigabyte GA-MA78G-DS3HP motherboard with a Phenom X4 9850. I am interested in upgrading to the Phenom II line, preferably the hexa-core ones, but budget constraints do not allow for a new motherboard. Would the X6 1055T work in my existing motherboard, and if it does, are there any measurable performance benefits with AM3 motherboards?
Here is the long and short of your situation: Wait until you can buy a motherboard to match a new CPU.
That being said, let's look at the details. The first being that AM3 chips do not work in AM2/AM2+ motherboards(source). The pin difference alone between the two parts make them incompatible.
As for the "performance gain" you can look at the passmark charts here. The X4 9850 gets a score of 2962 while the X6 1055T gets a score of 5193. That's roughly means that moving from an X4 system to an X6 system would boost your system performance by 75% (almost double). These numbers are of course an arbitrary unit of measurement and don't necessarily reflect actual perceivable performance gain of the upgrade. Every part of the system contributes to the overall system performance. Upgrading one component does have an affect of the system, but another component could limit how well your system performs.
It has been three months so the question may be irrelevant at this point. Let me give you some advice on the off chance you run into this situation again. When you look at building/upgrading a system, always save up so you can buy all of the parts at once. This allows you to match sockets and make sure everything is compatible with each other. When deciding what to buy, the general rule of thumb is that mid-range parts are generally the most cost effective. A good place to look for suggestions is the the PC perspective hardware leader boards. They will give you a good idea of what price point you should be at to get the most for your dollar.
Hope this helps