Does it make a difference whether I take my cymbalta (duloxetine) in the night or the morning?

Cymbalta (duloxetine) not really, can be a.m. or h.s. or twice daily, as tolerated.
No. Taking it at the same time every day is what matters. If a medication makes you sleepy, it's common sense to take it at bedtime, and if it makes you more awake, to take it in the morning.
At night. Most people who take Cymbalta (duloxetine) tend to have sleepiness as a side effect. Therefore, it is better taken in the evening. However, every person is different and some people get more activated by it and should take it in the mornings.
Morning is better. Taken in the evening, Cymbalta (duloxetine) can cause insomnia, so it is better to take it in the morning.