Concept:
The word isothermal means:
\[
\text{iso}=\text{same}, \qquad \text{thermal}=\text{temperature}
\]
Step 1: In an isothermal process, the temperature remains constant throughout the process.
\[
T=\text{constant}
\]
Step 2: For an ideal gas, internal energy depends only on temperature.
So, in an isothermal process for an ideal gas, internal energy also remains constant.
Step 3: But the defining condition of an isothermal process is:
\[
\boxed{\text{Temperature remains constant}}
\]
Therefore, the correct answer is:
\[
\boxed{\text{Temperature of the system remains constant}}
\]