I have a friend who has written a series of great nursery rhymes. I would like to put these rhymes into a book as a gift for my friend, but I don't want to give up my friend's first publication rights. Is there a service I could use — perhaps like one of those photo publishing books or something along those lines — where I could get the rhymes in a bound book form but without it being considered publication?
If you have art to go with these nursery rhymes, then I'd say go the photobook route already mentioned.
If not, check out using Amazon's Create Space. It's their print on demand service, that people can use to self publish. However, there's nothing saying that you have to offer it for sale -- you can set it up in Create Space, and print out a grand total of one, and have it shipped to you. And it'll only cost you a few bucks for the printing.
Another option may be a place that has an Espresso Book Machine, if there's one near you. The espresso book machine prints out library quality books (paper back), in a matter of minutes, on demand. It's like a giant copier but better. I say "may" because whoever owns it near you may not offer it for people to print their own stuff. The one near me does, and also offers help laying it out and making a cover on a per-hour fee basis. (it does have to be laid out exactly)
Finally, on Neil Fien's excellent idea in the comments of crafting the book by hand. You may not have the time or the tools to do that, but art or graphic design students at a nearby university do. I'm guessing you could commission one of them to do it for a couple hundred bucks, and get something pretty sweet if you make it clear you're commissioning an artwork, not asking them to slap some pages together. It gets even better if it's hand-lettered, not printed.
To track that down, I'd suggest find some classes on binding, materials (paper), or on fonts, and ask the professors for recommendations. Heck, if you're loaded and willing to spend the money, I bet you could get one of the professors would do it.
The more I think about it, the more I like the hand-crafted, hand-lettered idea. That would be a stunning gift!
Is the purpose of doing this for your friend to see their words "in print", or for them to see that someone considers them good? (i.e., you.) Or perhaps it's something else entirely.
Whichever option you choose, you'll have to get the rhymes in shape for printing/publication, if they aren't already. There's no need to hire an editor, not with a print run of one, but I would do some careful proofreading.
However, print runs of nursery rhymes will include illustrations as a matter of course. If you want to simulate a print run, you'll want to do these as well.
You have a few options:
Get the rhymes into bound book form, using a short-run print service.
Do a craft-style "printing" of the book.
This would be where you get a scrapbook from a crafts store, and use it to hold the pages. Call it a "mockup" of the book.
Do a framed print of one of the rhymes.
This is the simplest way to handle this. Pick one of the rhymes, have it illustrated or decorated, and print it out nicely and put it in a frame.