Trouble reading LM35 with long (?) Ethernet connection
The LM35 has trouble driving any kind of capacitive load, so the long runs of cable are generally not a great idea if you are relying on a voltage signal. The datasheet for it (The one I have is from the Texas Instruments website), has a few strategies to reduce this problem somewhat.

The approach that I have used successfully, is to use a current signal, rather than a voltage signal. This has the advantage that you only need a two wire twisted pair cable (I made my own twisted pairs - it is easy for relatively short runs).

The idea is that you connect the LM35 output to the ground pin with a known resistor value (the datasheet example has a 200 Ohm resistor). Since the voltage is 10mv per degree, the current wil be 10/R mA per degree. There will be an additional small current from the LM35 device itself, but this is small, and almost constant. Then connect the supply and ground to one end of your twisted pair.

At the other end, it is necassary to convert the current signal back to a voltage so that your Arduino can read it. Simply put a resistor (again, 200 Ohms is that used in the datasheet example) between the low side of the twisted pair to OV, the high side connected to 5V, and measure the voltage across this resistor.

Figure 17 in the TI datasheet is the one to follow.

Other things that help to get more stable, acurate readings;

Choose a time to take readings when you are not running anything else that might interfere (wireless comms, ultrasonic sensors, motors, solenoids, etc).

Take several readings. Ignore the first two.

Take the median value of the remaining readings (eg take 5 readings, ignore the first two, and sort the others in order. Then take the middle value.).

Hope this helps.