Since computer monitors are much more sophisticated than they were when that "web safe" distinction was made, you can choose other, more subtle colors for your site with confidence.
The next challenge is finding the codes for the colors . . ..
Kuler/Adobe Color CC: My favorite source of color inspiration was Kuler, which is now Adobe Color CC. It used to feature saved color palettes and you could create your own. But it still lets you find subtle colors and provides the hex code for using in your Weebly site. Just know that you will need to add a hashtag (#) before the hex code when you use it in Weebly (see below).
DigitalColor Meter: Macs come with a free app called DigitalColor Meter, which can be found in your Applications or Utilities folder. Under "View" select "Display Values" and choose the "Hexadecimal" setting. Use its eyedropper tool to sample colors you might like to use. You can sample from any image on your desktop, including images you have on your site and anything you can view on a browser.
The hex code will appear in DigitalColor Meter's three windows. Copy down the number pairs in order (R first, G next, etc.) and add a hashtag (#) at the beginning when you enter the code into the white window on the swatch palette.