I am looking to generate a small analog signal around -0.08V to +0.08V. What is the best way to go about doing this?
How would I go about generating a very small analog signal
By starting with a larger one and attenuating it. You can create such signals directly, but then you're operating closer to the noise floor and offset voltage of whatever parts you are using to create the signal.
A simple way to attenuate a analog signal is with something called a resistor divider. Use that as a search term and you'll find plenty of information on it.
You can use resistive dividers over a wide range of frequencies and you can control the input and output impedances by using T-networks.
A simple two resistor divider may suit- for example 990 ohms:10 ohms will knock an 8V signal down to 80mV with ~10 ohm output impedance. An advantage of this over active schemes is that offset voltage is less of a potential problem- 0.1% of 80mV is 80uV- which is a large thermal EMF for a resistor to develop but not much for an op-amp to contribute. Take care of the layout if you do this with low value resistors like 10 ohms because the copper traces can add significant resistance that will vary with temperature.
Finally, you can an attenuator by using a resistive voltage divider (say 99K to 1K ohms) and buffer it with an op-amp voltage follower. A good op-amp might have an offset voltage of 10uV or less, and the output impedance can be close to zero for low frequencies (and for current within the capabilities of the op-amp). This scheme requires dual supplies for the op-amp.