For new clients who want to create a new website on Weebly, I'm now going to ask them if being found through searches (like Google searches) is important. If it is, I'm going to recommend that they avoid the free website plan (mysite.weebly.com) and purchase a domain and a paid plan instead (mysite.com).
According to Weebly's own guide and confirmed by other research I've read, a website built on someone else's domain (in this case, Weebly's) is not going to do as well in searches. Here's the relevant section from the Weebly document about Domains:
Free Subdomains vs. Custom Domains
SEO is certainly something to consider if you're designing your own site. Not everyone needs to have excellent search results; if they're already well-known or they're just starting out and not ready to really "go into business," then the SEO results may not be a big deal. But now that I know this, I'll make sure clients know about this tradeoff if they want to go with a free starter website on Weebly.