Happily, that's easy to check. The Social Security Administration provides datasets of the number of newly registered children each year with any given first name. (It's how I know that "Philip," the proper spelling of Philip, suddenly became less popular than "Phillip," the incorrect spelling, around 1960. But that's an aside.)
Scanning a list of the names of presidents' children, I picked out names that were relatively unique and identified the year at which the father of each rose to national attention -- usually the year prior to running for president. And then I plotted the results.
"Sasha" (short for Natasha) saw a little spike in 2009 (the blue line at the bottom of that graph), but "Malia" actually spiked a little more, climbing from a few hundred kids in 1997 to almost 1,700 in 2009. Several names saw long downward slides over the course of their fathers' presidencies: Anna (FDR's daughter) and Patricia (Nixon's) among them. But two names stand out here: Amy, as in Amy Carter, and Chelsea, as in you-know-who.
A quick aside on Patricia Nixon. The graph above doesn't really do her name justice, since saying Nixon gained national attention in 1967 ignores his hard-fought 1960 presidential race and, of course, his tenure as vice president under Eisenhower. If you isolate the Nixon girls, the graph looks like this.
Bad timing for Patricia. Back to the overall popularity.
So let's say that Amy and Chelsea are the two names that spiked the most dramatically when their fathers became nationally famous. That's true, but it's also a bit misleading. After all, Amy was already a pretty popular name. Chelsea, on the other hand, rose out of nowhere.
So we took each name and figured out how much larger it was each year than the overall minimum count. So if in one year 200 kids were named, say, "Kermit," and the next year 400 kids were, that's a 100 percent increase. But "Amy" going from 20,000 to 22,000 is only a 10 percent jump.
When we look at the numbers that way, Chelsea stands out.
And so do two of Teddy Roosevelt's kids: the aforementioned Kermit and Quentin. In 1899, fewer than five kids were named Kermit. By the time Teddy left office, over 100 were. That's not a lot of Kermits, but it's a big increase.
So we feel pretty comfortable awarding Chelsea Clinton the title of "first child's name that most heavily inspired other childrens' names."
And in case you're curious, Chelsea's new daughter, Charlotte, already has one of the trendiest names around, climbing from about 1,000 kids with that name when Clinton's father left office to probably over 10,000 this year. Meaning that when Bill Clinton dances with a root vegetable as First Husband, there will be plenty of other Charlottes around to make fun of her.