"Beetle"s right, some meters just don't do well at very low power levels. The solution to that is to just use a little more power, something around 5 watts, give or take a few. There's not really all that much danger of harming anything at that power level, depends on what you are running, of course, but typically 5 watts ain't gonna hurt 'nuthin'.
There should be no difference in SWR at 5 watts, or 50 watts, or 500 watts, if the equipment used can handle that power level. SWR is a ratio, so it stays the same unless that ratio changes. Meters are not all created 'equal', and don't always give the same accuracy in readings at different regions in their scales. There shouldn't be really significant differences though. A few 'points' difference in the 'tenths' position are just not that significant. If the number in front of the decimal place changes, then that is significant and something just "ain't right". For those meters that require calibrating, recalibrating when changing power levels is a smart thing to do.
Having said all that, I'll add this to complicate things just a bit. SWR readings are one of the least useful things in knowing how well an antenna system if working. Some antennas with 'not good' SWR do much better than some antennas that are absolutely 'flat' (1.0:1).
- 'Doc