So, what are "best practices" here? I do a great deal of interactive PDF forms design & localization work, and right now I would say that your best move is to think about moving away from interactive PDFs entirely. I usually suggest a workflow that produces both interactive HTML-based forms and print forms, and I try to minimize the desirability of interactive PDF forms to the clients I have who ask me for strategic forms-management advice.
However, you might be required to produce interactive PDF as a target format with no say in the matter. (Happens to me all the time!) Plain-vanilla PDFs with interactive fields set up in InDesign won't do what you ask for. However, once you've set up your dropdown menu from within InDesign (actually called a "Combo Box"), you can open the Forms tools in Acrobat Pro and open up the Properties of the field in question. In the "Options" tab, there's a checkbox for "Allow users to enter custom text." This will allow the end user of the form to enter text from the keyboard.
Note that it allows the end user to enter free text in any field; it's not like obsoleted tools like Livecycle Designer where you could have nine immutable options and a tenth "Other" option with free text entry for the user. If that's doable in current Acrobat, I don't know anything about it. You can do a great deal with Javascript in Acrobat, though, so I don't discount the possibility that it could be doable, but I don't know how to do it.
(If I could learn how, I would rather spend that time learning about improving forms interactivity with JS in the browser, not in Acrobat.)
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