Is this something that would stabilize with time, like over many successive reads? Is there any good solution to this problem? Can I run the conversion slower? Is it as simple as just waiting a little longer between reads?
As you may know the arduino analogRead() function is designed and assumes the source impedance of voltage to be read is 10K ohms of lower. The fact that the AVR adc has but a single sample and hold capacitor but multiple multiplexed analog input pins means that you have to deal with the problem that trying to read a high impedance signal. As already posted you can lower the impedance by buffering the signal with a op-amp voltage follower. Or if you are only using a single analog input pin in your project you could just do multiple readings and only use the last reading. Hanging a cap on the analog input pin in question to ground can be effective in many cases. The best solution is only apparent in the context and requirements of the complete project, the purpose of the measurement, costs, power source used, and on and on. If this a personal 'hobby project' you are free to try and use simple or low cost solutions (that may not be optimum but still effective) that one may not be allowed or wish to use in a commercial professional setting.
So try stuff and see what works for you.
Lefty