I will guess that your grandmother is not much of a videophile, eh? I would suggest looking into cheaper brands such as Westinghouse, RCA or TCL (or one of the other cheaper, non-name-brand names). This will allow for large sizes without busting the bank. For $500, your budget is pretty much in line with name-brand 42 inchers; LG, Vizio, Samsung, and Panasonic all make 40-42 inch tvs for under $500. Picture quality will most likely be similar for the most part.
You might be able to find a 50 inch plasma (only 720p, though) for under $400 if you are patient.
It all depends if you want name brand or not, and if size is more important to your grandmother (no pun intended).