Hello there! The sentence "a friend of mine" is commonly used to initiate a narritive such as "A friend of mine once went to Paris on a vacation trip" or "A friend of mine really likes that singer!" when you're talking about things related to said matter. So, if someone you're speaking to talks about Paris, for example, you can say "A friend of mine once went to Paris on a vacation trip". Both "my friend" and "a friend of mine" could be used in that context and be correct. But if you'd use "My friend once went to Paris on a vacation trip" can leave the receiver of the message waiting for additional info, it makes your sentence seemed unfinished. I hope your question was answered and if you have any doubts, please don't hesitate to ask me!