Should Fama-French coefficients be calculated with daily or monthly returns?

In this context, I refer to the book on Empirical Asset Pricing by Bali, Engle and Murray (2016). They state on page 124 that

A minimum number of data points are usually required to ensure the quality of the values estimated by the regression. In the case of daily data over a one-year period, a reasonable requirement may be that the regression be fit using at least 200 data points. While using one year’s worth of daily data to calculate beta is common, other estimation period lengths and data frequencies are also used. Another common approach is to use monthly excess return data from the past five years.

Its basically a regression and usual regression thumb rules are applicable.

Since more data makes statistics more reliable, go with daily returns. If you're worried about noise, then you can smooth your returns with a moving average.

I'll add that daily data is not high frequency data. That term generally implies individual orders, or at worst the NBBO.