Concept:
In lanthanides, due to lanthanide contraction, ionic size decreases from La to Lu.
As size decreases, covalent character increases and basic strength of hydroxides decreases.
Step 1: Recall the trend of lanthanide hydroxides.
Basic strength decreases from:
\[
\text{La(OH)}_3 \to \text{Lu(OH)}_3
\]
Step 2: Identify the weakest base.
Since basicity decreases across the series, the weakest base is:
\[
\text{Lu(OH)}_3
\]
Hence, the correct answer is:
\[
\boxed{(C)\ \text{Lu(OH)}_3}
\]