Is it better to use a smaller, more accurate measuring cylinder several times or a larger, less accurate one for the same volume?
You have three types of errors:

The errors in the accuracy of your measurement.  Assuming you are reading the water levels consistently (at eye level at the bottom of the meniscus), these should be random.
To add random, uncorrelated errors, the standard method is to sum their squares and take the square root.  So, for the 0.1 mL average error for the 10 mL cylinder:
$$\text{average error for seven combined measurements} = \sqrt{7  \times (0.1 \text{ mL})^2} = 0.26 \text{ mL}$$
As you correctly intuited, it wouldn't make sense to simply multiply the 0.1 mL by 7, since this would be saying you have either a +0.1 mL error every time, or a –0.1 mL error every time, but that's not how random errors work—they are just as likely to be positive as negative. In addition, they don't all have a magnitude of 0.1 mL; rather, their magnitudes follow a distribution. Thus essentially what you have is a random walk, with a random distribution of step sizes, where the average step size is 0.1. The square-root-of-the-sum-of-the-squares method gives you the average distance you'd end up from your starting point with 7 of these steps

Error in the calibration of the cylinder.   This would be a systematic error, and it would add to the error of each measurement.  Higher qualities of cylinders (e.g., Class A glassware) have lower errors.

Error in the consistency with which the cylinder delivers the measured amount.  These cylinders are probably calibrated as &quot;TD&quot;, which means &quot;to deliver&quot;.  That is, it is assumed not all the water will drain out, so the markings on the cylinder are calibrated to account for that.  However, there will be some random error in how closely the amount that drains out corresponds to the assumed amount that drains out.  The only way to determine this error would be to contact the manufacturer (or to test it yourself using a scale).  In addition, if your cylinder is not clean, more water may adhere to the walls, which could cause a systematic error in the amount of water delivered.


Note also the comment by Andrew Morton that there's an additional important practical consideration, which is that if you have to use seven measurements, you could miscount and thus end up being off by  $\pm$10 mL! And this error is much easier to make when measuring with a graduated cylinder than, say, a tablespoon, because each measurement with the cylinder takes focus to get the total volume to 10.0 mL, and that focus can cause you to lose count.  There are of course ways to address this, e.g., making a mark on a piece of paper for each 10.0 mL added.