signal analysis - 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.