Some problems in genealogy can be solved using a single method or data source. For instance, if your friend's biological father happens to have been tested on the same DNA site as your friend was, then it will be trivial to tell who he is. However, as you've found, many genealogy problems require more information.
A principle of genealogy is to start with what you know, and work backwards to what you don't know, by deciding what data sources might be available that can help you, and then looking for those data sources. So you presumably know your friend's birth date and place, and her original birth name. From that, she should be able to request her birth certificate, if she doesn't already have it, and perhaps that will contain her father's birth name (probably not, but there's a chance and it should be easy to track down). You could also have your friend talk to people who knew her mother back then, or have been close to her since then, since they might know where she was living when she got pregnant, who she was associating with at the time, and so on; knowing that would help you narrow down people who could be the father. You don't mention any marriages, but it's possible your friend's mother was married at one time, in which case it should be possible to find a record (again, probably not, but it's possible).
As far as DNA, there are several things to consider:
First, if you find a very close DNA match then that might find the father. To maximize the chances of that, you should get your friend's DNA information onto as many sites as possible. Many sites allow you to upload your data from eg Ancestry DNA for free (as of now, I believe at least gedmatch.com, Family Tree DNA, and MyHeritage allow free uploads of data from another site). Beyond that, your friend might end up having to pay for tests from additional testing companies, just so that her results can get onto those other sites.
Second, as people say, having a known maternal ancestor tested would be useful, since it should let you eliminate many relatives who are on the mother's side.
Third, if you find a match who really is a second cousin (the estimates on sites are sometimes exaggerated, so that a person they say is a second cousin is actually a more distant relative), that person would have to have done a fair bit of genealogy research to know all the people who might be the connection. (It's possible that your friend's parents were part of a close-knit community of people who had lived in the same sparsely populated area for a long time, but probably not; if they were, then even a fairly distant relative might know who the father could be.)
I've heard it said that when you're using DNA you should also be going back and forth to traditional documents, and that may apply in this case as well. Obviously if your friend's mother had only a brief connection with the biological father, then there won't be records tying them together, but you might get lucky.
To summarize, since you haven't found an answer using the data source you have, expand your search, and try to combine the pieces of information you get from various sources. Good luck!