Several things spring to mind here.
The "brain dead" way to estimate things would be to take the mean of all the samples and divide by the standard deviation. But as you point out, that gives you a rather large standard deviation and it does not take advantage of everything you know. In general, such an approach would suffer very badly from aliasing: if you don't sample an exact number of wave forms, you will get huge errors in estimating the DC offset (in the extreme case imaging you are sampling just half a wave form and you can see what I mean).
Usually, if the frequency is not exactly known (or not an exact multiple of the sampling frequency) you can address this issue with a windowing function (Hamming or Hanning window) which give less weight to the samples near the ends of the sampling interval, and which results in less spectral spreading. But in this case you are interested in the mean (DC, frequency 0) of the signal, so I would suggest another approach.
First - find the amplitude of the sinusoid. If you are sure that your signal frequency is exactly locked to the sampling frequency, you can do a simple DFT. In Matlab, you could do:
t = 0:9999; % not interested in actual frequency; just need 10k samples
s = sin(2*pi*t/1000); % template of sin wave - 10 complete cycles
c = cos(2*pi*t/1000); % template of cos wave
normVal = s*s'; % value you would get for an amplitude of 1
s_component = signal*s'/normVal; % multiply and add - assume signal is 10k element row vector
c_component = signal*c'/normVal; % ditto
best_fit = s * s_component + c * c_component;
residual = signal - best_fit;
Now you can process the residual: you can see if there are any other frequencies present (for example you could do an FFT), and/or you can look at the RMS value of the mean.
It's usually a good idea to study residuals carefully: it tends to show you structure that you might not see when the main signal is on top of it. So compute it, plot it, histogram it, analyze it (do a normality test) and decide how to proceed. There is no 'one size fits all' answer.
As always - ask questions if you need clarification.