stevenindra - I viewed an online video of this experiment and several things come to mind that might contribute to the breakage problem. First, the jar that covers the 'filament' is not evacuated or filled with an inert gas. It is filled with ambient air, so as the filament becomes near white hot, it is combining with oxygen creating oxides that may alter the mechanical structure of the pencil refill. Second, placing the pencil refill into the alligator clips is actually placing mechanical stress on the refill, which is a fragile material. Those stresses may contribute to the refill breaking. Third, the same alligator clips are relatively rigid, so as the pencil refill expands significantly, additional stresses are introduced and when rapidly cooled (power is removed) the refill may break.
You might substitute flexible wires to attach to the refill then connect the wires to the rigid alligator clips which might remove much of the mechanical and expansion stresses. You might also find a way to slowly bring down the current level from 8 cells to 7 then 6 etc. Perhaps a rotary switch will allow you to switch quickly and minimize the mechanical shock on the refill. Last, but not least, the refill is a combination of graphite and inert fillers, usually clay formulations. I would try the softest refill you can obtain such as a 2B or 4B if they can be found. They will not be as mechanically strong as the more standard HB or #2 refill, but will have a higher concentration of graphite. On the other hand, harder refills such as 2H and 4H have higher electrical resistance so it contributes to the heating and brilliance of the filament. Any of these suggestions may improve performance. See attached link for some refill information. What started as a simple demonstration can be a basis for many variants that could make for an even more interesting experiment.http://pencils.com/hb-graphite-grading-scale/