Hang on (sorry, I need this to be clarified, I hope I haven't been doing something fundamentally wrong on what should be a simple question). I'll outline my reasoning in full.
So we started off with 0.1373g of sample, being heated and stuff and then put into a 500ml volumetric flask.
So we have 0.1373g in 500mL. Then we take a 1 ml aliquot. So we have less than 0.1373g in 1mL.
We then dilute this to 250mL. Our mass of sample from the 1mL aliquot does not change. Our concentration does, but that's not mass. We then analyse using AAS for how much Mercury there is in this 250mL.
So we have a
concentration level of 0.35 micrograms per litre of mercury in 250mL of solution. But what actual mass of mercury do we have in this 250 mL of solution? We have 0.0875 micrograms of Mercury.
We obtained this 250mL solution by diluting a 1mL solution. When you dilute, you don't change the amount of subtance you have. The concentration does change. (

, the volume increases and hence concentration decreases when you dilute something). Also note that m=nM, if the amount of moles stays the same, the mass stays the same.
So in the 1mL solution, we have still 0.0875 micrograms of Mercury, but in a higher concentration.
This 1mL solution was obtained from taking an aliquot of a 500mL solution. When we take an aliquot, you do change the amount of substance you have (

, the volume decreases AND the amount decreases by the same factor, hence concentration remains the same when you take an aliquot). The concentration of the 1mL solution and 500mL solution remains the same, but the amount of mass will be different.
This means we must calculate the mass of Mercury in the 500ml solution. 0.0875 micrograms * 500 = 43.75 micrograms of Mercury in 500mL.
So we can now say we have 43.75 micrograms of Mercury in 0.1373 g of sample. The answer wanted it in milligrams, so let us convert/apply sig figs: 43.75 micrograms = 0.04375 milligrams = 0.044 milligrams of Mercury in 0.1373 g of sample