Alternatives to "Good Night" when sleeping in the afternoon

It's a bit odd. I would have expected to hear "have a nice nap", as someone who's going to sleep at 1:00 pm is probably only going to sleep for an hour or two.

If you planned to sleep for seven or eight hours, starting at 1:00 p.m., that's longer than a nap. I guess in that case "sleep well" (or "rest well", or "sleep tight", if you prefer) would be more appropriate. "Sweet dreams" would technically work, but it's not usually used except with people you are very familiar with (children, romantic partners, family, etc.). You could also use "sleep well" at night — it's not restricted to napping or sleeping for long periods during the day.

Also, you can use "good night" for situations other than going to sleep. I often hear it used when people go their separate ways after spending some time at night together, they say "good night" as they are parting.

Yes, it is a bit odd. "Sleep well" or similar would be more appropriate.

I doubt you can find an expression which could cover an afternoon nap (of any length) and be similar to "good night".

Although it's true that, as FrustratedWithFormsD correctly pointed out, "good night" may be used for reasons other than wishing "a good night's sleep", it is also true that normally people sleep at night and for this reason we have a specific expression which also covers this activity.

Personally, I would go with "sleep well or "have a nice nap" (to my ear, "sleep tight" sounds more suitable for a longer sleeping period).

"Good night" in that situation is correct usage in the US.

Because most people don't go to sleep when the sun sets, instead going to sleep hours later (or near sunrise for college students..), the phrase "good night" has a colloquial meaning more along the lines of "have a good sleep", but it is shorter and easier to say.

It's used in informal settings regardless of the time of day, and mostly regardless of the length of sleep, if you plan on sleeping for more than around an hour.