Maybe? I mean, it somewhat depends on a number of factors: first, how well known is the New World outside of the Gaelic and Norse spheres? Obviously, people know that something is out there as there is trade, but that doesn't mean they know a lot. And I suspect that much of what they know is a mix of fact and a hazy cloud of tall tales and outright fabrications. It also depends on how secretive the Norse and Gaels have been about the New World; after all, they really don't want anyone edging into their market.
Secondly, what are the opportunities that second sons currently have in Europe at this time. Not having an inheritance (or much of one) is bad, yes. But that can be mitigated by other opportunities to make a name for one's self closer to home. Can they attach themselves to another Lord who can grant them land and protection? Are there border regions that might allow a well trained, if unlanded, member of the nobility to make a name for themselves and gain land and title? Does the Church offer a career path that might be a draw to some?
Finally, what reception could they hope for in the New World? Would the Ostish and Gaels look kindly upon some unlanded sons of Frankish nobility - who can't even speak the language - showing up to offer their services? Possibly. Or they might show them the door right quick too. Would the Native peoples be receptive to taking such men on? Once again, maybe - as long as the language barrier can be overcome and the Native peoples see a use for them.
My guess is that there could potentially be a trickle of such men coming to the New World. But not that many. Even with meager inheritances, there are a lot of opportunities closer to home that don't require a long and dangerous ocean voyage into a semi-civilized land (my apologize: I certainly don't think of North America in this TL or OTL as uncivilized at this point in history, but our hypothetical Franks likely would) which may or may not have many opportunities available. There would certainly be some who are interested and excited by this type of scenario, but I think they will decidedly be a small minority..
I’m just looking forward to when people other than the Irish or Ostish find out that North America exists.
Besides, even a trickle of Franks would be interesting. After all, I don’t think any of the European groups in North America came in what we’d call large numbers.