Ways to analyze electrical signals without FFT?
Please be specific about what kind of analysis you want to perform. My reply is based on the assumption that you want to measure the exact frequency of the input (since you are talking about frequency resolution) - that is known to be near 15 Hz.

Averaging is not the only option to remove the noise. You can filter your input signal by a low-pass filter with a cutoff-frequency near the frequency of interest. If your noise is very spiky (just a glitchy sample with an abnormally high amplitude), there are non-linear techniques which might be even more efficient at removing spurious transient events without much impact on the overall shape of the signal - for example, median filtering. The danger of averaging is that the averaged segments will have to be perfectly synchronized with the period of your input signal - if they are not, the phase differences will cause cancellations and your signal will lose amplitude as segments are averaged.

Even with averaging and/or in a situation with access to a limited number of samples, there are better techniques than the FFT. For example, accurately determining the frequency of a sinusoidal signal can be seen as a model fitting problem and can be solved by least square techniques. You could also use a PLL to try to lock into the input signal, and thus get the phase/frequency information - if this is what you are interested in knowing...