2015-related question: why are Lucas-Carmichael numbers named after Lucas? Summary
2015 is a so called Lucas-Carmichael number. I believe (for reasons that I will explain below) that the 'Carmichael' in the name is a reference to ordinary Carmichael numbers and not to the person Carmichael. (When ignoring  that the Carmichael numbers are of course named after the person Carmichael.) So the question arises how 'Lucas' entered the name, especially since he was dead for some years when Carmichael numbers (let alone Lucas-Carmichael numbers) were defined. Does anybody know when Lucas-Carmichael numbers were so named and why?
Background: Carmichael numbers
From Fermat's little theorem one can cook up a nice test to see if a number $n$ is prime: choose a number $a$ and see if $n$ divides $a^n - a$. If the test says 'no', then the answer [to the question 'is $n$ prime?'] is 'no' as well. If the test says 'yes' then, hmm, we don't really know what the real answer is. We have some more faith that $n$ might be prime, but to be more certain we better try some other value of $a$. Now this naturally leads to the question: are there numbers $n$ such that $n$ divides $a^n - a$ for every $a$ coprime to $n$ but with $n$ nevertheless not prime. Needless to say it would be really annoying if such numbers exist.
Unfortunately they do: Carmichael found the first example (561 = 3 * 11 * 17) in 1910 (according to Wikipedia). In 1994 it was even proved that there are infinitely many.
More background: Korset's criterion
Some ten years before any Carmichael number was known, Korset proved (perhaps in an attempt to rule out their existence?) that a number $n$ is a Carmichael number if and only if $n$ is square free and
$$p-1|n-1 \textrm{ whenever } p|n$$
with $p$ prime.
Now this doesn't do much to improve the usefulness of the Fermat primality test, since finding all prime factors of $n$ is, trivially, at least as hard as testing whether $n$ is prime, but it is still a very nice result.
Lucas-Carmichael numbers
Lucas-Carmichael numbers are defined as squarefree numbers $n$ satisfying
$$p+1|n+1 \textrm{ whenever } p|n$$
with $p$ prime.
(E.g. 2015 = 5*13*31 and 2016 = 6*336 = 14*144 = 32*63)
So it looks very much like they were invented when someone thought 'hey, let's see what happens if we replace all the minuses in Korset's criterion by pluses!' This is a legitimate thing to do in situations like this, as is naming the resulting numbers after the Carmichael numbers they were (seemingly) inspired by. 
Only I would opt for something like 'topsy-turvy Carmichael numbers', 'Ultra-positive Carmichael numbers', 'anti-Carmichael numbers' or something along these lines. In other words, the question remains:
What do Lucas-Carmichael numbers have to do with Lucas?
 A: The earliest primality tests were based on Fermat's Little Theorem, according to which $a^{p-1} \equiv 1 \bmod p$ for primes $p \nmid a$, which deals with the multiplicative group of the finite field with $p$ elements. Later, Lucas discovered tests that were based on what we know as a natural subgroup of order $p+1$ inside the multiplicative group of order $p^2-1$ in the finite field with $p^2$ elements. This justifies naming objects you get by replacing elements of order dividing $p-1$ by elements whose order divides $p+1$ after Lucas, even though he did not have to do anything with the objects in question, in this case Carmichael numbers.
The first Carmichael numbers, by the way, were found long before Korselt found his criterion by the Czech priest Vaclav Simerka in 1885; see my article here.
A: As you already mentioned, a Carmichael number is a composite number $n$, that passes Fermat's Little Theorem for every integer $a$, which is relatively prime to $n$, i.e. $a^{n-1} \equiv 1 \mod n$. Fermat's Little Theorem could also be seen as a (not so reliable) probabilistic primality test (due to the existence of infinite Carmichael numbers). Korslet showed that a composite integer $n$ is a Carmichael number if and only if $n$ is square-free and for all prime divisors $p$ of $n$ it holds that $p-1 \mid n-1$, which is also known als Korslet's criterion.
Why are now Lucas-Carmichael numbers called like that? It certainly has to do with Lucas sequences, which could also be seen as a generalisation of the Fibonacci sequence. Let $D, P$ and $Q$ be integers such that $D = P^2-4Q$ is non-zero and $P>0$.  Let $U_0(P,Q)=0$ and $U_1(P,Q)=1$. The Lucas sequence of first kind associated with the parameters $P, Q, D$ is defined recursively for $n\geq 2$ by $ U_n(P,Q)=PU_{n-1}(P,Q)-QU_{n-2}(P,Q).$
There are also congruences that hold for primes based on the Lucas sequences. One of them is the following congruence:
Theorem 1: Let $p$ be an odd prime such that $\gcd(p,QD)=1$. Then it holds that $U_{p-(\frac{D}{p})}\equiv 0 \mod p$, where $(\frac{D}{p})\in \{-1,0,1\}$ is the Jacobi-symbol.
This congruence can be used as a probabilistic primality test, which we call the Lucas primality test. Similar to Fermat's little theorem, where Carmichael numbers defeat this primality test, there are also composites that defeat the Lucas primality test. Exactly because of this reason, they are called Lucas-Carmichael numbers.
Definition: Let $D$ be a fixed integer. A Lucas-Carmichael number is a composite number $n$, relatively prime to $2D$, such that for all integers $P,Q$ with $\gcd(P,Q)=1, D=P^2-4Q$ and $\gcd(n,QD)=1$, that passes the Lucas primality test.
Williams showed an analogous theorem to Korslet's criterion for Lucas-Carmichael numbers, which affirms the naming even more:
Theorem 2: Let $D$ be a fixed integer. Then $n$ passes Theorem 1 if and only if $n$ is square-free and $p-(\frac{D}{p}) \mid n - (\frac{D}{n})$ for every prime divisor $p$ of $n$.
If $(\frac{D}{p})=1$ for all primes $p \mid n$, then the Lucas test completely reduces to the test based on Fermat's little theorem.
If $(\frac{D}{n})=1$ it was shown that many composites that pass the Lucas primality test also fool the primality test based on Fermat's little theorem, hence we usually require for primality tests that $(\frac{D}{n})=-1$. (Similar arguments hold for the Miller-Rabin test and the strong Lucas test, which are stronger variants of the two primality tests.) This is a reason why often square-free composites $n$ with $p+1 \mid n+1$ for all primes $p$ dividing $n$ are just called Lucas-Carmichael numbers.
I hope this gives the answer you were looking for.:)
