According to everywhere I've looked, the 24C02S uses 3 address pins, statically hard wired to set the address of the chip. What if the 24C02S is in a Sot23-5 package? VCC, GND, WP, SCL, SDA are all I have to work with. Datasheets aren't helpful in what to do in this case. Is it a fixed address that can't be set? If so, is it 1010000 or 0x50?
24xx serial eeproms generally follow the addressing shown in this Microchip datasheet.
For SOT23-5 devices, A0..A2 are zero. LSB is ignored (as in right-shifted away) so the address is 0x50 for 1K and 2K devices. The 24C02 is a 2K device.
For larger devices the lower address bits are commandeered to select the memory page so the I2C address might be 0x50 or 0x51 for a 4K device, 0x50, 0x51, 0x52, 0x53 for an 8K device and 0x50..0x57 for a 16K device.
If any of the address lines are changed 0x50 will no longer work assuming they are address lines. However I do not believe they are address lines. You need VCC, Gnd, SCL, SDA, to make it work on I2C. That is 4 pins minimum leaving two more. These can be: 2 address lines, 1 address line and WP, or It is also possible one of the pins is a NC (No Connect) A lot of information on these can be found in the Arduino forums. You can also find schematics and code that will you to test them. My guess is it is a "5-pin" (in 6 pin package), Pinning out 6 pins even if one is a NC makes the board much more useful. AT24C02 which is a 2K SERIAL EEPROM. Internally it is organized with 32 pages of 8 bytes each. You will have to use an an 8-bit data word address for random word addressing. Is it possible to show a section of the schematic that has this part.