For a very good, very traditional version try this one
It may be old-fashioned, but all the important, traditional nursery rhymes are on it, and sung extremely well by the children's choir of a radio station (nearly 40 years ago so it's not exactly hip - but they do sing in tune).
You can listen to many songs on this CD online if you follow the link to decide if you or the prospective recipient might like it.
My 2-year-old and even my older ones like the songs and I'm glad he does/they do, as I can't stand the more modern versions.
However there is no specific book for this CD.
If this traditional stuff is not quite your cup of tea and you want something many people in Germany buy for their children nowadays (although many still use the traditional songs), type in "Ritter Rost" which is not too bad, or "Rolf Zuckowski" which I personally dislike, but many children and even some parents swear by it (some contain educational politically correct messages, eg songs about fastening your seatbelt in the car and stuff like that).
Books:
We have this one if you want to try to get it elsewhere, it's unfortunately no longer available on amazon.
It's complete with piano accompaniment for each song but I reckon you might want something simpler anyway, more for the child directly.
We also have a thinner, more child-friendly songbook but unfortunately not even a picture is available on amazon.de.
But this one
contains loads of traditional German songs. I don't have it but I've seen it and it's very nice.
How old is the child? If he's roughly between 2 and 4 years old there's something completely different which he might like, called "Bobo Siebenschläfer". Even my 5 and 7 year olds, who've really outgrown it, sometimes still demand it occasionally.
These are everyday stories about a humanised dormouse toddler and his families. Both books and CDs are available, and I have yet to come across a child who doesn't like Bobo Siebenschläfer. Better check with his parents though he doesn't have the books/CDs already.
There are 3 different books and CDs available in various editions.
Bobo Siebenschläfer on amazon.de
I'm German so if I can be of any further assistance let me know.