To monitor magnetic field fluctuations you will need a Hall effect sensor. In my experiments I have used a Honeywell SS494B Hall effect sensor, which is available online for $3-$4. Other sensors should work also, just pick an analog model, not latching, with comparable or better sensitivity. The Honeywell SS494B promises to be sensitive enough to provide around 5mV per 1 gauss. For a frame of reference, the earth's natural magnetic field measures around 0.5 gauss, a refrigerator magnet is about 50 gauss, and a neodymium magnet is in the 1000's of gauss.
The Honeywell SS494B is a very sensitive Hall effect sensor, as far as low cost hall effect sensors go, but its measuring range still reaches over 400 gauss. Depending on the construction of your water meter enclosure, if it's made of metal or plastic, the magnetic field strength outside the meter can be as low as 1 gauss or less. This presents a challenge, since 1 gauss or less is at the extreme low end of the Hall effect sensor's measuring range.
But before tackling the challenge of measuring very small magnetic fields, let's take a look at how the Honeywell SS494B Hall effect sensor works. The sensor has 3 pins: power (Vcc), ground, and output. If you were to look at the output from this Honewell Hall effect sensor, simply powered from a +5v source, you would see that the output sits at around +2.5v, or roughly half of Vcc. This is called the quiescent output voltage, or in other words, this is the voltage the Hall effect sensor will output when no magnetic field is present. But if you hold a magnet in front of the sensor, the magnetic field will pull the output voltage either towards ground(0V) or Vcc(5v), depending on the polarity of the magnetic field.
The tricky part comes next. If you were to measure field strengths of 1 gauss or less, the Honeywell SS494B will deviate no more than ~5mV from the quiescent output voltage. So, to use an Arduino to measure such tiny voltage fluctuations, and with good resolution, amplification will be required.
To amplify the signal coming from the Hall effect sensor, a general purpose operational amplifier like the LM324 will work just fine. In my experiments I have used a widely available and cheap (less than $0.25) LM324 op-amp, and the schematic in the next step is based on the LM324 op-amp.