How to add an edit to a profile

It depends on your access-rights.
If outside your personal family-group you probably have to invite one of the profile managers to collaborate with you to get full access.

When looking in Tree View - is there a blue border around the Profile Photo area?
If so, that is a claimed profile and only the person themselves can edit it.

The only way to "add an "edit" option" is to contact the person and convince them to go in and change their Account Settings to allow everyone in their family group to edit their profile

To get to Account Settings - click on your name at top of page, just to right of profile photo - to right of search box. Then -- -- Account Settings -> Permissions -- then click on box in front of "Let my family edit my profile"
You cannot do this except for your own Profile. And a person cannot just give you (or some other specific person(s)) that right - it is reserve right to edit their own profile to just themselves or let all family members edit their profile.

There is however another reason why you cannot see the edit button: You have a language setting where the translated text does not give room for the edit button. Not necessary visually, but it have to be enabled by Geni staff (for Norwegian we had to ask about this even if visually the Norwegian word for tree is shorter than the English.

Try changing to English (US) and see if the option is visible again.
BTW: I assume you have the default tree-view.