For very high bandwidth systems the use of a 1-bit analog-to-digital converter is a promising solution to keep the power consumption of terahertz receivers manageable. With 1-bit quantization being a highly non-linear operation, it is necessary to revise all signal processing algorithms, including synchronization algorithms. In this paper we focus on the estimation of an unknown deterministic frequency offset between the transmitter and the receiver. To this end, two different data-aided frequency offset estimators are presented which both operate with 1-bit quantization at the receiver and in the presence of additive white Gaussian noise, and which correspond to the existing weighted phase averager and the planar filtered estimator used in conventional receivers. Comparing the performance of the considered estimators, we find an interesting trade-off between the achievable estimation performance, the computational complexity, and the estimation range. The planar filtered estimator has a higher complexity and achieves theoretical performance bounds for low signal-to-noise ratios in comparison to the weighted phase averager. Performance at high signal-to-noise ratios is limited by an error floor for both estimators, which can be lowered by oversampling in the case of the planar filtered estimator.