I have a Swedish friend who is really interested in Scandinavian history, religion/mythology and old literature (she is studying Old Norse). She knows a fair bit about Vikings and their history, but I was thinking about giving her a book about the topic. It could be pretty 'advanced' (she already knows far more than the basics) and even focusing on a specific topic. Indeed, a book covering unique aspects of this part of Scandinavian history might be the best choice!
I'd recommend Kristina Ekero Eriksson, Gamla Uppsala : människor och makter i högarnas skugga. (publisher's page) It's a popular-archaeological account of the results from the 2012-2013 excavations in Old Uppsala, which are the largest ever performed at the site. (well-known from the Sagas) The formal archaelogical report of the dig came out last year and this summary came out just this year, so it's not so likely she's read it already.
Other than that, here's for instance the mandatory reading list for Uppsala University's course 'Vikings and the Viking Age':
Jesch, Judith, Women in the Viking age. Woodbridge: Boydell, 1991
Jones, Gwyn A history of the Vikings 2. ed., [reissued]: Oxford: Oxford University Press, 2001, cop. 1984
Näsström, Britt-Mari, Fornskandinavisk religion : en grundbok. 2. uppl.: Lund: Studentlitteratur, 2002
Sanmark, Alexandra, Power and conversion : a comparative study of Christianization in Scandinavia , Uppsala: Dept. of Archaeology and Ancient History, 2004
Sawyer, Birgit; Sawyer, Peter, Medieval Scandinavia : from conversion to Reformation, circa 800-1500 , Minneapolis, Minn.: Univ. of Minnesota Press, 1993
Snorre Sturlasson, d ca 1179-1241; Hollander, Lee M. Heimskringla : history of the kings of Norway Austin: 1964
Snorri Sturluson, Werenskiold, Erik; Johansson, Karl G. Nordiska kungasagor. : 2 Olav den heliges saga , Stockholm: Fabel, 1993
Out of these, she certainly has Snorri already if she's studying Old Norse. Gwyn is one of the most popular introductory books. This is an introductory course, so if by 'advanced' you mean that she's studied the subject at the university level, then she may be past all this.
For a more 'advanced' student of the Vikings, I might recommend Neil Price's The Viking Way. It's a really interesting pairing of Old Norse sources and archaeology to explore the shared mindset of viking violence and seidr worldviews. I also like Lotte Hedeager's Iron Age Myth and Materiality which also tries to square Old Norse sources with earlier material culture. I think both are advanced but still accessible even to non-specialists, and if your friend is taking coursework in Old Norse, either might be a nice complement to her course of study.
Asking for resources is fine here. A quick glance at the medieval section of the subreddit books list turns this up:
Brink, Stefan, ed. The Viking World (2008). A book which combines many short chapters on any topic relevant to Vikings or the Scandinavian late Iron Age. A strong point is that many chapters are written by the relevant specialists instead of a single author who is trying to specialize in everything. A bad point is that this means that there's not much of a central theme connecting the chapters, which makes this more of a reference work than a bedtime story.