Large sample rate or large number of cycles when estimating frequency? I wish to determine the frequency, amplitude and phase of a sine wave with some noise and I can only have a fixed number of samples. Should I go for many cycles and a small sample rate or a few cycles and a large sample rate?

I know that if I choose to use a DFT to determine the frequency then the spectral resolution will be the sample rate divided by the number of points. I must avoid aliasing so this suggests I should choose a small sample rate giving for example 3 or 4 points per cycle and capture many cycles of data. It may be a separate issue but the actual frequency will probably not correspond to a line in the spectrum so there will still be the, I think, unresolved issue of finding the frequency amplitude and phase from the small cluster of points around the maximum in the spectrum. 

I could also choose an alternative method where I curve fit a sine wave to the data. This should give me best estimates in the least squares sense. However, for this method it is not clear which is best many cycles with a small sample rate or fewer with a large sample rate.

Is this a known problem with a known solution? Thanks for any input.
For frequency estimation of a pure sine wave, the difference between 2 slightly different frequency sine waves increases over time.  So a longer time base (as long as the sample rate is well separated from the Nyquist rate, or any integer sub-multiple thereof) would provide a bigger difference in a fixed number of samples, thus making it more likely that two different frequency signals could be separated from the effects of noise on any one signal.  This bigger separation difference is a close equivalent to better frequency estimation accuracy.  Note that under-sampling is possible if the S/N ratio is good enough, and as long as you know, a-priori, the correct frequency range of your signal and stay away from integer multiples of Fs/2 being near to Festimated.

A longer time base also corresponds to a narrower frequency bin spacing of a DFT result, thus decreasing the percentage errors in any interpolation between bins that may be performed or required.  But any frequency is a line in the spectrum after (high-quality/Sinc) interpolation.

Once you estimate the frequency, you can use two or more points (depending on the S/N ratio) to least squares fit a sine wave equation in two unknowns to estimate amplitude and phase.  Or windowed-Sinc interpolation, followed by successive approximation is another method to estimate amplitude.  Quadrature modulation of the signal against the estimated frequency can also be used to estimate phase (most accurately when referenced to the point or time at the center of the data window).