Assuming we have no synchronization with the source or knowledge of the actual channel over the time of acquisition: the ability to acquire the sinusoid, and the signal to noise ratio of that resulting acquisition, is completely limited by the frequency dispersion of the sinusoid as received (due to the relative phase noise and frequency stability at the transmitter and receiver), the power spectral density of the noise at the receiver, and the inevitable spurious (noise tones) that will exist in addition to the white thermal noise floor, within the bandwidth of that "spread" sinusoid as received.
For this a 20 bit ADC is not really helpful: the quantization noise of the ADC as a noise density in a good receiver with appropriate front-end gain, will be significantly below the receiver front-end noise floor. However it can facilitate shifting receiver functionality such as gain control from the analog to the digital (so consideration to high number of bits is driven by the dynamic range requirements- what is the ratio of strongest signal to minimum discernable signal and how much of that gain control (AGC) is done in the analog?), but that isn't the driver here for optimizing the minimum discernable signal. A target minimum SNR must be specified (this could be derived directly from the minimum acceptable RMS error on the estimate of the amplitude for example), and from that the minimum number of bits is derived with consideration to sampling rate and the averaging details further described below. Any additional bits can then be used in place of analog gain control for varying signal levels.
What is needed is good clocks (transmitter local oscillator, receiver local oscillator and sampling clocks) and an understanding of the temporal changes in the channel. Understanding clearly the relative frequency stability between transmitter and receiver if very helpful in optimizing reception when total observation time is not limited: for short term stability we use to phase noise, and for long term stability if the system / electronics allow sufficient stability, we use the Allan Deviation. Both factors (same underlying process which is the measure of stability of the transmitted sinusoid, as received) effect ultimately how long can we observe a sinusoid to improve the SNR by effectively filtering with a bandpass filter centered on the sinusoid's frequency - the FFT is one example of implementing such a band pass filter.
The signal as received, once effected by the transmitter's clock, receiver's clock and channel variations and dispersion will inevitably drift over time. Since the assumption is the signal is weak ("buried in the noise"), we have no way to actively track that drift in order to remove it (synchronization)- so this will ultimately limit the observation time for which the sinusoid will stay within our bandpass filter. The bandpass filter is necessary to remove the noise, everywhere except where the sinusoid occupies. Consider an extreme (and unfeasible) condition where we send a perfect sinusoid (no phase noise, frequency offsets or drift) over a stationary channel and receive with an equally perfect clock and our receiver noise is strictly amplified white thermal noise (no spurs). Under that condition we can observe as long as we want, and noting that filter bandwidth is the reciprocal of averaging time (averaging is a filter), we can reduce the total noise within the bandwidth that gets narrower and narrower as a trade with observation time. The reality is the system will not be stationary, ultimately phase noise (which becomes in the longer term frequency drift) and channel variations over time will limit the blind observation time. Spurious signals which can be very low will at some point compete with the detection of the sinusoid (but that can also be addressed with good frequency planning in the receiver and not using any signals where intermodulation products can create tones in receiver bands of interest).
That said, my approach would be to review the frequency stability of the transmitter and receiver (improving the sources used to the extent possible/ feasible) as well as the expected temporal variations of the channel. With those parameters understood the Allan Deviation can be used if stability allows for observation times exceeding 1 second and longer. If the system exists, then captures could be made of actual received sinusoids (in much higher SNR conditions, which means closer to the transmitter), and with that data the Allan Deviation can be computed. The floor in the Allan Deviation curve would indicate the maximum observation time for which the SNR can be optimized through effective filtering. This would thoroughly characterize the transmitter and receiver and some of the channel (as the distance between transmitter and receiver increases, the effects of the channel will inevitably increase and will decrease the maximum possible observation time).
The filtering done is correlation with the sine wave specifically, which is a band pass filter and optimum under white noise conditions - this is exactly what we get with an unwindowed FFT with the sine wave on a bin center. However given it is a sine wave, I recommend a simple product and integrate rather than FFT (this is essentially 1 bin of the FFT with fine tuning control). If the frequency is not known an FFT approach may be of interest for "rapid" acquisition (the FFT duration here is still that optimum observation time determined, so rapid in the sense that we can probe multiple possible frequency locations at the same time), and then once acquired (tracking mode) switch the the "one bin" approach with fine control to increase SNR if possible. The signal will inevitably occupy a bandwidth over the time observed, and the equivalent noise bandwidth of the filter when optimized will match this bandwidth and thus provide the maximum signal power compared to the noise power due to the noise density over that same bandwidth.
