Lean amine acid gas loading, lean amine temperature, and absorber pressure are more likely to have a noticeable impact on H2S level in your treated gas, i.e. lower lean H2S loading, lower temperature, and higher pressure all favor H2S absorption. Higher wt % solution will give you more moles of amine per unit volume of solution circulated. If you run high rich loadings and cannot circulate more amine for some reason the higher solution strengths will help the rich loading. I doubt you'll see much of an impact on treated gas H2S concentration vs. amine strength unless you're running your rich loadings on the high side.
Higher solution strength also tends to increase hydrocarbon solubility in your solvent, particularly troublesome BTEX components if they are present in the gas stream you're treating. Its been my experience that higher strength amine solutions have higher foaming tendency and foam stability in the amine regenerator too, particularly with higher molecular weight amines such as MDEA.
There are a lot of technical articles available on amine systems here:
The Laurance Reid CDROMS have a lot of good amine unit information if you have access to them as well..
Do you have enough instrumentation available to know the temperature profile in your absorber? If not, climbing the tower and shooting temps at inspection ports can get the job done too. Knowing the temperature profile and how many "cool" trays you have can be particularly useful for troubleshooting absorbers.
What kind of gas are your treating, is it Claus tail gas, recycle hydrogen, sales gas? The type of system can have a big impact on potential problems and solutions.