I'm not sure if this category has a particular name - usually until someone cares enough to give one of these a name or nice notation they just have long unwieldy names. I can suggest some notation though - if your algebra is $A$ then I think $$D^{-,b}_{\mathrm{fd}}(A)$$ is pretty much standard - the fd for finite dimensional maybe not so much but conditions on the cohomology traditionally go there. If want the components finite dimensional as well then you could drop the fd and use $A$-$\mathrm{mod}$ instead of $A$. You probably already know all of this though...

Hopefully there is a name for this I just don't know about and someone can tell you.

Post Coffee Edit: Actually the above is slightly silly - it is perfectly resonable to consider the homotopy category of bounded above complexes (usually of projectives) with bounded finite dimensional cohomology but in the derived case you may as well just look at $D_{\mathrm{fd}}(A)$. Unless you have a particularly good reason for wanting bounded above complexes you may as well take all isomorphs in $D(A)$ - this is correct philosophically. You can then drop the bounded (since finite dimensional total cohomology implies boundedness) and just call it the derived category of $A$-modules with finite dimensional total cohomology.

I'm not sure if this category has a particular name - usually until someone cares enough to give one of these a name or nice notation they just have long unwieldy names. I can suggest some notation though - if your algebra is $A$ then I think $$D^{-,b}_{\mathrm{fd}}(A)$$ is pretty much standard - the fd for finite dimensional maybe not so much but conditions on the cohomology traditionally go there. If want the components finite dimensional as well then you could drop the fd and use $A$-$\mathrm{mod}$ instead of $A$. You probably already know all of this though...

Post Coffee Edit: Actually the above is slightly silly - it is perfectly resonable to consider the homotopy category of bounded above complexes (usually of projectives) with bounded finite dimensional cohomology but in the derived case you may as well just look at $D_{\mathrm{fd}}(A)$. Unless you have a particularly good reason for wanting bounded above complexes you may as well take all isomorphs in $D(A)$ - this is correct philosophically. You can then drop the bounded (since finite dimensional total cohomology implies boundedness) and just call it the derived category of $A$-modules with finite dimensional total cohomology.

I'm not sure if this category has a particular name - usually until someone cares enough to give one of these a name or nice notation they just have long unwieldy names. I can suggest some notation though - if your algebra is $A$ then I think $$D^{-,b}_{\mathrm{fd}}(A)$$ is pretty much standard - the fd for finite dimensional maybe not so much but conditions on the cohomology traditionally go there. If want the components finite dimensional as well then you could drop the fd and use $A$-$\mathrm{mod}$ instead of $A$. You probably already know all of this though...

Hopefully there is a name for this I just don't know about and someone can tell you.

I'm not sure if this category has a particular name - usually until someone cares enough to give one of these a name or nice notation they just have long unwieldy names. I can suggest some notation though - if your algebra is $A$ then I think $$D^{-,b}_{\mathrm{fd}}(A)$$ is pretty much standard - the fd for finite dimensional maybe not so much but conditions on the cohomology traditionally go there. If want the components finite dimensional as well then you could drop the fd and use $A$-$\mathrm{mod}$ instead of $A$. You probably already know all of this though...

Hopefully there is a name for this I just don't know about and someone can tell you.

Post Coffee Edit: Actually the above is slightly silly - it is perfectly resonable to consider the homotopy category of bounded above complexes (usually of projectives) with bounded finite dimensional cohomology but in the derived case you may as well just look at $D_{\mathrm{fd}}(A)$. Unless you have a particularly good reason for wanting bounded above complexes you may as well take all isomorphs in $D(A)$ - this is correct philosophically. You can then drop the bounded (since finite dimensional total cohomology implies boundedness) and just call it the derived category of $A$-modules with finite dimensional total cohomology.