My mom recently signed up for FamilySearch. However, I had created an entry for her years ago. How do I merge the entry I made with my mom's new account?
Short answer - you don't.
Your mother will have created a profile for herself, as the root person of her new account. Only she can see that profile, for reasons of personal privacy. You created a profile for your mother that only you can see for the same reason.
So I understand it correctly, the solution is to have two records of the same person on FamilySearch? It doesn't make any logical sense to me that there is no way for my mother and I to work from the same family tree.
That's how the system is designed. When someone passes, and the death information is added to the various copies of the profiles, then we merge them.
There is a project in the works - but not quite ready for prime time - for family groups to work together on living profiles. There will be a presentation about it at RootsTech this year.
Personally, I don't add living profiles to the FSFT. And, you and your mother can work together on the deceased profiles in your family.
@DavidSouthworth4 You can still work on the same tree, your deceased ancestors can (and should) be linked. You just won't be able to see the living people that she adds, and vice versa.
Edit: I didn't notice until after posting this that @Áine Ní Donnghaile already explained that you could still work on deceased ancestors together. For some reason I just blipped over the last sentence.