The following hints may help you to deal effectively with grade disputes:
Ask students to define their grievances and express their justifications in writing before they come to your office. You may want to make your grievance policy explicit before any disputes arise by stating it explicitly either in your first day handout or in the paper assignment.

It is perfectly reasonable to ask the professor or another TA for the class to read the assignment and give it a grade. Agree beforehand that both you and the student will abide by the third person's decision. (Advise the student that his/her grade may go down rather than up in this situation.)

Remember, and remind students, that you do not "give" grades; they earn them.

You may want to give them the option of rewriting. Be sure that your policy on rewrites is in accordance with the professor's and the other TAs'.

Remember, and remind students, that your concern is with the particular assignment and grade at hand, not with their overall GPA, scholarship or program eligibility, etc.

Remember you have the prerogative to take a matter under consideration; you are not obligated to respond immediately in defense of your judgment. In fact, some TAs have a policy of refusing to commit to a grade change until they have had a chance to look at the assignment at home; this time allows you to consider the matter without an anxious (or, in some cases, belligerent) student in your presence.

Finally, remember that some cases may indeed warrant an alteration of a previously assigned grade. New TAs may wish to discuss such cases with fellow TAs or faculty instructors before doing so.
If students are not satisfied with your assessment of their work or resolution of other conflicts, they should go to the professor. The undergraduate academic program advisor, the chair of the undergraduate committee, and the department chair also may be asked to assist in resolving difficult conflicts. Authority for arbitration lies first with the faculty instructor and next with the Dean of the College of Letters and Science.

For more information, see the EDKB guide to Grievance Procedures.