In QGIS 3.x (and QGIS 2.18 as well) you can easily configure different symbol sizes in your legend via expressions based sizing.
If "Legend" is the Item ID of your legend use the following expression for your symbol size (i.e. 5 in legend, 10 in canvas/map):
if(@item_id = 'Legend',5,10)
Accepted answer works in QGIS 2.0.
For 3.0, you must invoke setActive on the rule
rules = layer.render().rootRule().children()
As far as I know there is still no solution to adjust the symbol size of point features inside the QGIS composer. The symbol entry works only for polygons and lines.
However, a workaround that was not mentioned yet is to export the map as an svg-file and open it with Inkscape for example (Inkscape is open source). There you can manually adjust the symbol ...
ArcMap has the Cartographic Representations tool that allows you to create your own symbology with tons of options.
Check it out: https://desktop.arcgis.com/en/arcmap/10.3/map/working-with-layers/introduction-to-the-cartographic-representations-tutorial.htm
One workaround is to use a geometry generator style.
Under the Marker symbology, change Symbol layer type to 'Geometry generator', select Point/Multipoint as the Geometry type, then paste in an expression like translate($geometry, 0.2, -0.2) to offset the point (change the numeric values as needed, these are X and Y offsets in the CRS of the layer), then ...
When I have some funky symbology like this I usually create a folder in my table of contents called "Legend Layers", in that I create duplicates of my layers which have symbology that doesn't look good in the legend. I use these duplicates to change the symbology to something that looks good in the legend and then I add those to the legend instead of the ...