Having taught this before, I understand that this can be a little tricky to get your head around at first, but it requires less mental gymnastics than you think. When it says 1 mole LiH, it is talking about the compound as a unit. Consider a single molecule of H2O. If you took that single water molecule and broke it into its constituent parts, you wouldn't have (as per your logic) a third of an atom of oxygen and two thirds of an atom of hydrogen - that doesn't make any physical sense. Instead you have one atom of oxygen and 2 of hydrogen.
You have to look at it as a ratio of atoms per compound, and remember that a mole is a unit of molecules. IOW, what is true for 1 molecule is true for 6.02 x 1023 molecules. In your case, for every 1 molecule of LiH, you have 1 lithium and 1 hydrogen. You can expand that to moles, and say that for every 1 mole of LiH, you have 1 mole of lithium and 1 of hydrogen. In my water example, for every 1 mole of H2O, we have 2 moles of hydrogen and 1 of oxygen. In a more complicated example, say Mg(NO3)2, for every 1 mole of Mg(NO3)2, you have 1 mole of Mg, 2 moles of N, and 6 moles of oxygen.