Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.

*Question 1*: What class of lattices the following property characterizes? $$\mu(\hat{0},a)=\mu(\hat{0},\hat{1})\mu(a,\hat{1}), \ \forall a \in L$$ It follows that $\mu(\hat{0},\hat{1}) = \pm1$.

*Remark*: It is satisfied by any boolean lattice, more generally by the face lattice of any convex polytope (as suggested by John Shareshian), and more generally by any [Eulerian lattice][1].    
*Proof*: An Eulerian lattice is a graded lattice $L$ such that for any $a,b \in L$ with $a \le b$, we have $\mu(a,b)= (-1)^{|b|-|a|} $, with $a \mapsto |a|$ the rank function. The result is immediate. $\square$   

*Question 2*: Is there a non-Eulerian lattice with the above property on the Möbius function?  
**Yes**, see the answer of John Machacek.  

As suggested by Sam Hopkins:    
*Question 3*: Is there a non-Eulerian *atomistic* lattice with the above property on the Möbius function?  
*No* for $|L| \le 12$, as checked by the following Sage program (using [these lists](http://www.shsu.edu/mem037/Lattices.html) of Martin Malandro):

    from itertools import product
    def relationtest(L,n):
    	for l in L:
    		P=Poset((range(n),l))
    		b = P.bottom()
    		t = P.top()
    		if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
    			L=LatticePoset(P)
    			if L.is_atomic():
    				if not L.is_graded():
    					print(P.cover_relations())
    				if L.is_graded():
    					for x, y in product(P, P):
    						if P.compare_elements(x,y)==-1:
    							if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
    								print(P.cover_relations())
    								break

 

  [1]: https://en.wikipedia.org/wiki/Eulerian_poset