What is the quantity 2(handles)+crosscaps called? It is well-known that up to homeomorphism, the complete set of orientable surfaces is $\lbrace S_g : g=0,1,\dots \rbrace$, where $S_g$ is the sphere with $g$ handles.  The complete set of non-orientable surfaces is $\lbrace N_k : k=1,2, \dots \rbrace$, where $N_k$ is the sphere with $k$ crosscaps.  
Typically the genus of $S_g$ is defined to by $g$, and the genus or sometimes non-orientable genus of $N_k$ is defined to be $k$.  I would actually prefer to define the genus of $S_g$ to be $2g$ and the genus of $N_k$ to be $k$.  In some sense this is more natural since if $S_{g,k}$ is a sphere with $g$ handles and $k\geq 1$ crosscaps, then $S_{g,k} \cong N_{2g+k}$.  Moreover, I am writing a proof where I want to proceed by induction on some sort of genus.  It seems more natural that my invariant should go down by 1 when moving from $S_{g,1}$ to $S_{g,0}$ instead of going down by $g+1$.  
Question. What is the name for the invariant $S_{g,k} \mapsto 2g+k$?
 A: You could certainly call it the Betti number $b_1$ without bringing in homology. 
A: I don't know of a name for the specific quantity $2g + k$, but it should be noted that $2 - (2g+k) = \chi(S_{g,k})$, the Euler characteristic.
A: My colleague, Chris Cooper, has some unpublished notes in which he calls this quantity the $\it weight$ of the surface. I'm not aware of anyone else using this terminology. 
We use these notes for our Topology lectures. The relevant chapter is available at http://web.science.mq.edu.au/~chris/topology/chap04.pdf
A: This is an eample, I believe, where the names started off on a bad track and prevented any better replacement from developing.
"Genus" even in the orientable case is often a cause for confusion, and it becomes really
terrible with the two kinds of genera.   So, people end up
using circumlocutions:  "the non-orientable surface of Euler charcteristic -2", etc., obscuring their geometrry
If all you want is the function, then $2 - \chi$ is perfectly good.
The trouble is that genus is often used as part of a naming convention,  emphasizing  the
description in terms of semigroup generators for surfaces under connected sum, that is, the torus and the projective plane.
In many cases where these quantities are important,  2-dimensional orbifolds are also important.   Surfaces with boundary, and surfaces with points removed, , are also important.  The semigroup of 2-dimensional orbifolds is not finitely generated, although generators are easily parametrized.   I struggled with getting an appropriate way of talking about these simple objects that had complicated names
them for a long time, in particular in discussions with Conway.   When I convinced him (talking in terms of all these kinds of
circumlocutions) that
orbifolds are a good way to think about symmetry groups, he had exactly the right instinct ---
he came up with a good systematic notation,.   Oriented surfaces are (O^n) for
the surface of genus n and $(X^n)$  for the unoriented surface of unoriented genus n.  You can read about the rest
of the notation elsewhere.   Again, the function $2-\chi$ doesn't require another name.  Just the surfaces and orbifolds require better names.
Conway has advocated the terminology "double torus"   "triple torus"   and "n-tuple torus"
rather than 2-holed torus etc. which are terms of endless mixup, because people think of punching a holes until they've  had much
indoctrination.
