Assuming you have a led backlight, and the lcd is taking 5V power... Seems the lcd was already replaced once? And the backlight was not originally used by the main board, resulting in a hack.
RA was added on as a series resistor for the backlight, likely led. This was added because the voltage for the backlight is higher than the backlight wants (4.2V based on the LCD). It's 56Ω, so the current is likely around <30 milliamps? The data sheet says it normally takes 260mA, so I'm guessing the power supply is limited.
J2, the backlight Anode jumper makes me think the backlight is powered via the main power for the LCD, instead of Pin 15. That it has a solder blob and an smd resistor that didn't reflow properly says a lot about the hack that built this (no offense).
J16 is the backlight Cathode jumper. The left side goes to pin 16, the right is likely connected to the A pad on the left of the board. See feature 5 on the data sheet where it says LED can be driven by pin 1/2 (Main Ground and VDD), pin 15/16, or A/K. The wire likely connected to the center pin, a direct connection for ground to the backlight. The wire goes to a switch that disconnects ground to turn off the backlight, obviously.
To replace it with a new one, the data pins 1-14 will be the same. Since R8 is missing, you have to provide power via Pin 15 (or A) the Positive for the LED. Since the power supply is limited, you may want to add that 56Ω resistor in series or it will try to pull a few hundred milliamps. Since R7 is there (6.8Ω), the backlight may be on as soon as led + power and normal ground are connected. To use the switch, you will need to remove R7, and connect that switch to Pin 16 (or K).
A more detailed data sheet for a similar version of the ACM1602s confirms it. R7 ties it to normal ground (Pin 1), while R8 would have connected it to VDD (Pin 2), like your existing lcd uses. Instead of RL1 being the current limiting resistor, it has a 0 ohm jumper and uses R7 to limit the current, likely to 120 to 240 mA.
Side note, RL3 is useless. 1 megohm across the led for no reason? Weird.