Observationally, the X-ray spectrum (\(0.5-10\) keV) of low-level accreting neutron stars (NSs) can generally be well fitted by the model with two components, i.e, a thermal soft X-ray component plus a power-law component. Meanwhile, the fractional contribution of the power-law luminosity \(\eta\) (\(\eta\equiv L^{\rm power\ law}_{\rm 0.5-10\rm keV}/L_{\rm 0.5-10\rm keV}\)) varies with the X-ray luminosity \(L_{\rm 0.5-10\rm keV}\). In this paper, we systematically investigate the origin of such X-ray emission within the framework of the advection-dominated accretion flow (ADAF) around a weakly magnetized NS, in which the thermal soft X-ray component arises from the surface of the NS and the power-law component arises from the ADAF itself. We test the effects of the viscosity parameter \(\alpha\) in the ADAF and thermalized parameter \(f_{\rm th}\) (describing the fraction of the ADAF energy released at the surface of the NS as thermal emission) on the relation of \(\eta\) versus \(L_{\rm 0.5-10\rm keV}\). It is found that \(\eta\) is nearly a constant (\(\sim\) zero) with \(L_{\rm 0.5-10\rm keV}\) for different \(\alpha\) with \(f_{\rm th}=1\), which is inconsistent with observations. Meanwhile, it is found that a change of \(f_{\rm th}\) can significantly change the relation of \(\eta\) versus \(L_{\rm 0.5-10\rm keV}\). By comparing with a sample of non-pulsating NS-LMXBs probably dominated by low-level accretion onto NSs, it is found that a small value of \(f_{\rm th} \lesssim 0.1\) is needed to match the observed range of \(\eta \gtrsim 10\%\) in the diagram of \(\eta\) versus \(L_{\rm 0.5-10\rm keV}\). Finally, we argue that the small value of \(f_{\rm th} \lesssim 0.1\) implies that the radiative efficiency of NSs with an ADAF accretion may not be as high as the predicted result previously of \(\epsilon \sim {\dot M GM\over R_{*}}/{\dot M c^2}\sim 0.2\) despite the existence of the hard surface.