"a random day" is correct and usable in written English.
You can use it to describe a regular day that is unplanned or unexpected. For example, "I woke up this morning not knowing what I would do on this random day."
I chose the edition of the Guardian published on Saturday 25 September, a random day.
Researchers at Loyola University examined ninety-one Facebook pages seeking kidney donations on a random day in October, 2011.
Could Nemes have made the work more spare, choosing a random day on which nothing occurred but the ritual toil of horror, like the scrubbing of bloody floors?
A skim down my own friends' updates (names changed, naturally) on a random day yielded plenty of less-than-scintillating information ("Sam Hopkins is watching The Thing and folding laundry") but also more promising material.
By enquiring into burden on the last day without headache, we assumed this was a random day for each participant.
The odds of having headache a random day decreased during 22 weeks by 0.65 (0.50-0.84 0.50-0.84CI)).
Sometimes, when Nate is making his regular love to Jennifer, with her smooth skin and soft belly and perfectly black hair, his mind wanders back to the old days, when a random Thursday night might have delivered him a freckled brunette in a scoop-necked shirt to make love to — where?
I had a look at Instagram on a random Monday at 5 30pm, just as the Museum of Modern Art in New York closed for the day.
But still, I think, it's nice to be here, all together, on a random Sunday.
"It was the last thing I expected on a random Tuesday morning".
It would be a lark to go to see a bagpipe band together on a random Tuesday in March.
