This is admittedly not well-explained in the show.
From Rip's comments, it seems as though they need to make a big change to the timeline to prevent Savage's rise1 (emphasis mine):
Rip: I also said that time wants to happen, and such a world-changing event like Savage's rise to power can't be stopped merely by kidnapping his young pawn.
Legends of Tomorrow Season 1 Episode 10: "Progeny"
They knew at the beginning that kidnapping the kid was a longshot:
Snart: Fine; if y'all don't have the guts to kill this kid...
Rip: Then removing him from the timeline might be the next best thing.
Legends of Tomorrow Season 1 Episode 10: "Progeny"
It's worth noting that Rip emphasizes the word "might" here.
I would point out that there is a significant difference between killing the kid and simply kidnapping him, especially if Tor Degaton (Per's father) is aware of his son's fate. Having the boy simply disappear, with no indication of what happened to him, creates a situation of unresolvable hope for Tor. We might imagine, for instance, that he begins to neglect (or embezzle) his company to search for his son, putting him in a vulnerable position for Savage to take over. Whereas, if he knows his son is dead, we might imagine him burying his grief with work, further cementing his status in the company and giving Savage no way to gain power.
Both situations are just speculation, of course, and we have no way of knowing how history would have actually played out in either situation; but it does serve to illustrate that kidnapping the kid and killing the kid do not, necessarily, have equivalent outcomes.
Since kidnapping Per Degaton evidently didn't have the desired affect, Rip appears to be hopeful that killing him would be a large enough disruption to knock time off-course.
It's entirely possible that it wouldn't have worked anyway, of course, but what have they lost for trying?
1 This is why they've been so focused on killing Savage; it's hard to think of a much bigger disruption than that