A “baby Christian” is a person who has recently converted to Christianity.
The imagery of new Christians as being “babies” in their faith hearkens back to the Bible itself, from passages such as 1 Corinthians 3:1-2 (“But I, brothers, could not address you as spiritual people, but as people of the flesh, as infants in Christ. I fed you with milk, not solid food, for you were not ready for it. And even now you are not yet ready.“) and 1 Peter 2:2 (“Like newborn infants, long for the pure spiritual milk, that by it you may grow up into salvation”).
Although this imagery of a new convert as a “baby” is as old as the Bible, the specific term “baby Christian” has flourished in the English language mainly since the 1970s. The graph below, courtesy of the Google Ngram Viewer, shows the relative popularity of the term “baby Christian” during the 20th century. As you can see, the term’s use rocketed during the 1970s and has continued to climb in popularity up through the present day.
The term baby Christian was discussed on the Dictionary of Christianese last summer when it appeared in an issue of Christianity Today. See that link for sample quotations and additional observations about its meaning.
One last point: don’t confuse baby Christians with cradle Christians. Cradle Christians are Christians who were born into Christian families and grew up going to church. Since cradle Christians have been Christians all their life, they almost always have long grown out of their “baby Christian” phase.