Large sample rate or large number of cycles when estimating frequency?
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).