Proposal

Please re-state the problem we are trying to solve

Clicking on the paypal payment option more than once causes the 'Edit' option to go away.

What is the root cause of the problem?

In BasePaymentsPage.js the edit option is only shown if the selected option is of type CONST.PAYMENT_METHODS.PAYPAL and when we click on it, the selected option is set to that. But when we click on it once again, onClose() callback for the popover containing the action buttons is also called.
https://github.com/Expensify/App/blob/5a1b35c297a6b9c6f2ee502a674b7f460bd90734/src/pages/settings/Payments/PaymentsPage/BasePaymentsPage.js#L417

And this onClose() then calls resetSelectedPaymentMethodsData, which clears out the selected account option (currently paypal) and sets it to null.
https://github.com/Expensify/App/blob/5a1b35c297a6b9c6f2ee502a674b7f460bd90734/src/pages/settings/Payments/PaymentsPage/BasePaymentsPage.js#L210-L218

What changes do you think should me made to fix this?

There is no need of clearing the selected option on closing the popover. So, we can just change the following line in Popover:
https://github.com/Expensify/App/blob/8a856e7adc5ff3a3c2e463371d6b4a8b42644e86/src/pages/settings/Payments/PaymentsPage/BasePaymentsPage.js#L417
to

onClose={() => hideDefaultDeleteMenu(false)

This will make sure that the menu is hidden but the selection is not reset because shouldClearSelectedData will be set to false

Now, there might be doubts as to the need of the selection reset code then... Well, the function to reset the selection resetSelectedPaymentMethodData was introduced in this PR: Expensify/App#14869
This PR was fixing the case of closing the modal if the payment option is deleted from another device. And where this code is essentially needed is in the useEffect that handles the change in the payment methods here:
https://github.com/Expensify/App/blob/8a856e7adc5ff3a3c2e463371d6b4a8b42644e86/src/pages/settings/Payments/PaymentsPage/BasePaymentsPage.js#L360-L365
Therefore, it still is fulfilling it's purpose and is being removed from where it is not needed. So, I think we can safely set shouldClearSelectedData to false when calling hideDefaultDeleteMenu() to prevent the running of the function resetSelectedPaymentMethodData()

What alternative options did you explore? (Optional)

from plantonic.