Noisy Frequency Estimation using Hilbert Transform
Digital Hilbert transform is tricky to do since an ideal Hilbert transformer requires an FIR filter with an impulse response from -infinity to +infinity. Obviously that can't be done, so numerical problems are unavoidable. They tend to be worst at low frequencies so my guess is you'd see a smaller error at, say, 100 Hz. Andrew Duncan's paper "The Analytic Impulse" is thorough analysis of what the digital Hilbert transform can and can't do. See http://www.andrewduncan.ws/air/index.html

Some alternative methods of measuring frequency have been discussed here: Is there an algorithm for finding a frequency without DFT or FFT?