The risk to the grass from mowing is that the grass blade once cut open is susceptible to infection by fungus. So what is really needed is for the grass to have a chance to heal as quickly as possible and for that it needs to be dry. So the best time would be late morning on a clear day, giving the dew a chance to clear and giving the grass the bulk of the day to dry off. Also, you don’t want to mow wet ground because the soil will become compacted, so water the day before, or if the weather is rainy, mow a day after it has stopped raining. The other things which one should keep in mind when mowing to reduce lawn stress are keep the blade sharp, which makes a clean, small wound to the blade (a dull blade will make a jagged rip) and to mow frequently enough so that no more than 1/3 of the blade is removed (sometimes this means mowing more than once a week, sometimes it means you can wait 2 weeks).
The same logic, BTW, applies to watering. You want to keep the grass wet for as short as possible because fungus likes water. So water deeply (most people don’t) and only when necessary, and do it in the morning (like 9-10 AM) on a clear day giving the grass the day to dry out (some people water early AM to avoid competition for water pressure or due to local ordinance but this leaves the grass wet longer). Sometimes I will water during a light rain or heavy dew since the grass is already wet. If the weather is extremely hot it is ok to water the lawn at midday to cool it off (11-12AM, usually against local ordinance) and doing this prevents some kinds of fungus.