Grad student asking department for increase in stipend
Yes, you can certainly ask that the department reconsider the amount of your stipend. However, you should also prepare for the case where they do not adjust it. 

Normally, I would start by talking with your advisor, as she will most likely have to be on board for any increase to be successful. Then, I'd approach either the graduate student advisor orthe graduate student administrator for the department. There may already be rules and policies in place for how to request a stipend increase. If that doesn't work, the next steps are probably to contact the chair of the department and the dean for graduate students (or other appropriate officer at your institution). 

However, depending on where your funding comes from, it may or may not be possible to get an increase. For instance, NSF or NIH stipends are normally set at a uniform level across a college or university. It's also possible that the students with higher stipends have outside funding which reduces the actual cost to the college. Finally, your advisor may not be able to afford a large stipend increase.