White Hairline Mark Around Artboard When Export

Nina, thin unwanted lines when exporting an image are usually caused by anti-aliasing.
When the dimensions of an object in Illustrator contain fractional pixels values, they are rounded to whole pixels (a fractional pixel does not exist).
If you want to export as an image, it is better to use pixels (or points) as units instead of centimeters or inches. You can set the units by Ctrl clicking the intersection of the Rulers
That way you can make sure that you don't have fractional pixels in your pattern tile, which can show as lines caused by anti-aliasing.
Double click your pattern in the Swatches panel. This will open the Pattern maker and you can make sure that your Width and Height don't contain fractional pixelvalues.
When you want to save a pattern tile, just drag the pattern swatch from the Swatches panel to an empty space on the artboard.
This should show as a group in the Layers panel.
At the bottom of the group there is an empty rectangle that defines the bounds of the pattern.
Select it and choose Object > Artboards > Fit to Selected Art.
Show Rulers and  double click their intersection to set the 0/0 point in the upper left corner, make sure that the rulers are Global Rulers (this is the case when it states "Change to Artboard Rulers").
Now you can export the tile as an image.
Make sure that Artboards is selected, use Art Optimized for Anti-aliasing and use multiples of 72 ppi (like 144, 288) for the resolution.