Either tense could be used with some minor rephrasing.
I was too late to meet the deadline. The lesson I learned from this was that I should have done it sooner.
I was too late to meet the deadline. The lesson I learn from this is that I should start things sooner.
One difference is whether you are going to put the lesson into the context of present behavior, or, in the case of the former example, if you just want to indicate that you have learned your lesson and leave it at that. Another is in the perspective of the second sentence- are you reflecting now on what lessons you had learned, or are you presently observing now what lessons can be learned from the event in the past?
