All the Nicknames Are Taken!
–Still Love Eleanor
Sure, you could use Laney as a nickname for Eleanor! Parents have lots of leeway when it comes to pairing up names and nicknames. I think a lot of people will really like the combination of old-fashioned Eleanor with more contemporary Laney.
You could consider Elle, Ella, Elsa, or Elsie as nicknames for Eleanor, but those might be too close to Ellie. Or you could use the full Eleanor and skip the nicknames altogether.
But as always, in the long run, it’s Eleanor herself who will be in charge of her nickname—or lack thereof. She might love Laney and reinforce your plan to use it. But she might also prefer Eleanor. Or she might start introducing herself as Ellie even if she does have a cousin by that name.
When your daughter is young, you'll be able to exert your influence over what she's called. Later, though, it will be more difficult. That’s why I usually advise parents to be prepared for anything. If you absolutely can't stand a nickname, then you might want to avoid its associated full name. If you are dead-set on a certain nickname, you might want to put it right on the birth certificate, rather than leaving it up for debate. Since neither of those is the case here, it looks like you've found an elegant solution to the Case of the Already-Taken Nicknames.