There are a total of eight lanthanide ions given in the question: La3+, Ce3+, Ce4+, Gd3+, Yb2+, Yb3+, Lu2+, and Lu3+. We need to determine the number of diamagnetic lanthanide ions among them.

Diamagnetic materials are those that do not have any unpaired electrons and are not attracted to a magnetic field. In other words, all the electrons in a diamagnetic material are paired up. On the other hand, paramagnetic materials have unpaired electrons and are attracted to a magnetic field.

To determine whether an ion is diamagnetic or not, we need to consider the electronic configuration of the ion. The electronic configuration of the lanthanide ions can be determined by knowing the number of electrons in the outermost energy level.

Let's analyze the electronic configurations of the given lanthanide ions:

1. La3+: [Xe] 5d0 6s2
2. Ce3+: [Xe] 4f1 5d0 6s2
3. Ce4+: [Xe] 4f0 5d0 6s2
4. Gd3+: [Xe] 4f7 5d0 6s2
5. Yb2+: [Xe] 4f14 5d0 6s2
6. Yb3+: [Xe] 4f13 5d0 6s2
7. Lu2+: [Xe] 4f14 5d0 6s2
8. Lu3+: [Xe] 4f14 5d0 6s0

Now, let's determine the number of unpaired electrons for each ion:

1. La3+: 0 unpaired electrons
2. Ce3+: 1 unpaired electron
3. Ce4+: 0 unpaired electrons
4. Gd3+: 7 unpaired electrons
5. Yb2+: 0 unpaired electrons
6. Yb3+: 1 unpaired electron
7. Lu2+: 0 unpaired electrons
8. Lu3+: 0 unpaired electrons

From the analysis, we can see that the ions La3+, Ce4+, Yb2+, and Lu2+ have all their electrons paired up, indicating that they are diamagnetic. Therefore, the correct answer is '4' - there are four diamagnetic lanthanide ions among the given options.