Well, she's still your best source of information, and any secondary sources (e. g. if you have aunts and uncles, or you contact her friends or classmates from that time) may very well tell her that you're doing this, or at least won't lie if confronted with a direct inquiry.
What we try to tell people in your situation is to come clean with your mother about the search but don't make it about parenting skills or even about wanting to find yourself. Instead, make it objective - make it about a concern about genetic diseases that you may have inherited.
If you absolutely oh my God I can never ever ever let her know
, well, if you do come in contact with your biological father, it's possible that he will contact her. Or your mother will notice at some point that you're spending time with someone who looks eerily familiar. But let's let that sleeping dog lie for the moment. Probably your best bets are either generalized searches, e. g. Google, Facebook, and even Twitter, and possibly grabbing a free trial and looking on Ancestry.com.
If his name is common, or it's similar to that of a celebrity, you are going to have to wade through a lot of extraneous noise. For example, over 35 years ago, I dated a guy named Paul Ryan. Because a guy with the same name ran for President, there is no way I can locate this ex-boyfriend. If I really, really had to locate him (I don't), I would do as I am suggesting you do, plus I would just keep refining my searches, e. g. Paul Ryan -Senator
Another option is, consider the town you are from, and where your mother is from if they differ. Did your biological father go to High School with your mother? If so, then start looking at reunion sites and groups on Facebook and classmates.com (they will charge you).
You can also pay for a skip tracer but they are not cheap. The one thing you have going for you in this search is that it is a lot less likely that he has changed his name over the years. But unfortunately, without involving your mother in your search, you are tying both hands behind your back in this search.
Also, just an FYI, your biological father isn't necessarily going to be too happy to see you. And, further, the guy who you think is your bio father might not be. Perhaps there is even a forgotten third party involved. Absent a DNA test with the putative biological father (if I were him, and were suddenly confronted with a brand-new insta daughter or son, I would insist on one), you don't really know.