What a timely and interesting project!
Bisphenol A is an aromatic molecule (it has ring-shaped structures of carbon atoms), which means it will absorb light. In theory, Bisphenol A (BPA) concentration can easily be measured used a spectrophotometer (an instrument that shines light on a sample and measures how much is blocked/absorbed by the sample). Many high school classrooms have access to one or several-- maybe even a middle school. Otherwise, they are mostly found in labs on the university campus. BPA absorbs light maximally at 211 nm and 276 nm (the near ultraviolet). This can be converted to concentration either by making a standard curve (use a known amount of BPA in several clean samples, then use those values to make a line of best fit to match your unknown samples against), or by using the extinction coefficient of BPA, a known value.

Unfortunately, many other substances (such as amino acids and full proteins, for instance) also absorb in this range. When scientists measure BPA, they typically use complex techniques such as high performance liquid chromatography (HPLC) to separate BPA from other substances before measurement. An HPLC is probably not available for a middle school project.

Confounding this, the amount of BPA that leeches out of plastic is very small-- in the parts per billion (the equivalent of finding a pinch of salt in an olymptic swimming pool), so extremely sensitive equipment is necessary.

I did a quick search to see what had been used, but there's a caution to that since most articles were about more complex samples (blood plasma, urine) than just water. In any event, it seems that at the least one would need (also due to the small amounts to be detected) to use a GC/MS machine.. In addition, there may be some pretreatment of the samples required (may not be absolutely required for just measuring BPA in water).

Some articles for future reference: