While I'm transcribing Dilworth's Catechism for the online documents project, I ran across the following: " ... I cannot help taking Notice of the ridiculous Names which are given to some Children, as Jack or Sam; Sue or Betty, because other Children of the same Parents, who have been named John or Samuel, Susanna or Elizabeth, have died when they were very young, or in their Infancy; by which Means this holy sacrament of Baptism ... is done by the Sponsors in such a shameful Manner, as gives a great deal of scandal to all good Christians ..." The date of publication is 1772, and so Dilworth's pot shot is aimed at what appears to be a practice in England at that time which was common enough for him to take a swat at it. I know some of my Roman friends who entered the Roman Church as adults tell me they took a new name (if they were baptized or re-baptized) that was different from the name they ordinarily use. I've not run across that practice among Anglicans (yet), but perhaps will do so shortly if anyones know of it who sees this topic and comments further. The sacrament of baptism aside, I've always felt pity for people whose parents stuck a nickname on them, rather than a more formal name. Formal names (Kathryn, for example) easily generate nicknames (e.g. Kathy), but the process does not run the other way. And, so, I'm grateful to have been named William, grateful that I was always Billy to Mom and Dad, and grateful that my flock can call me Father Bill. In North Texas, that appellation has a mildly crusty flavor to it which comes in handy in some situations.
At least naming your child a nickname is nowhere near as bad as some of the crazy names you can find these days. I think the "Christian name" practice started to decline when birth certificates started to be issued. Before, the closest thing to birth records were baptismal records. I'm not sure what to think of this replacement really. It's nice to have a specifically Christian name to go along with the newness of the Christian life. At the same time, I don't think such things are necessary and a great number of people in our society still have names with Christian origins anyway.
So what is the quote in the OP trying to say? That parents name their second kids after the first kids who had died before Baptism?
No, Spherelink; just the opposite. Evidently, in his day (when infant/child mortality was high) there were parents who would name children with diminuitive forms (nicknames) of their already deceased infants/children. Dilworth -- for reasons he doesn't spell out precisely -- finds this so deplorable that he brings it up when discussing names in connection with the section of the catechism dealing with baptism.
We can only speculate from Dilworth's comments, of course. He finds the practice deplorable, and it's likely because of its rationale in the minds of the sponsors. But, he never tells us what that rationale is. If it is what you suggest, I'd wager (from what Dilworth says generally about the significance of the baptismal name) that their practice diminishes (in Dilworth's estimation) the overall honor, seriousness, and significance of the sacrament. I don't read a lot in any literature of that era, but when/if I do, my antennae will now be tuned to pick up any references to this practice if I stumble across them.