How do I negotiate an offer for Ph.D. funding?
As long as you are polite and don't come with a list of unreasonable demands, you don't have to worry about seeming ungrateful. 


  How should I broach the subject of receiving more funding without coming across as ungrateful


Try this:

"When we spoke last weekend, I got the impression that the level of funding could be negotiable. Is that still the case? I appreciate the 50% TA position very much, but I'm concerned that it won't cover my living costs."


  Is there any way to know what kinds of additional aid the university might have available


Ask them:

"Are there any other kinds of aid (besides the TA position) that I could be considered for?"