Step 1: Recall what isoelectronic means. Two species are isoelectronic if they have the same number of electrons. For lanthanide ions, we count electrons by subtracting the ionic charge from the atomic number. Step 2: Examine option 1 - Pr$^{3+}$ and Nd$^{3+}$. Pr (Z = 59): $59 - 3 = 56$ electrons. Nd (Z = 60): $60 - 3 = 57$ electrons. Not equal, so not isoelectronic. Step 3: Examine option 2 - Tb$^{3+}$ and Dy$^{2+}$. Tb (Z = 65): $65 - 3 = 62$ electrons. Dy (Z = 66): $66 - 2 = 64$ electrons. Not equal, so not isoelectronic. Step 4: Examine option 3 - Eu$^{2+}$ and Gd$^{3+}$. Eu (Z = 63): $63 - 2 = 61$ electrons. Gd (Z = 64): $64 - 3 = 61$ electrons. Both have 61 electrons! They are isoelectronic. Step 5: Examine option 4 - Pr$^{3+}$ and Ce$^{4+}$. Pr (Z = 59): $59 - 3 = 56$ electrons. Ce (Z = 58): $58 - 4 = 54$ electrons. Not equal, so not isoelectronic. Step 6: State the final answer. Eu$^{2+}$ and Gd$^{3+}$ both have 61 electrons and form the isoelectronic pair, corresponding to option 3. \[ \boxed{Eu^{2+} \text{ and } Gd^{3+} \text{ are isoelectronic}} \]