First of all, you need to verify if you have the same problem as me. Install some other free payment module like Skrill or Stripe (no need to configure it) and then go to back office, Design -> Positions and in the searchbar type Payment. PayPal and Skrill should be in the same hooks, especially in paymentOptions hook. If Skrill is there and PayPal isnt, this is a problem. In front office Skrill would work properly and show up in checkout while PayPal wouldn't.
First you need to get access to PhpMyAdmin. Very often it is in cPanel or other panel that comes with your hosting.
In phpmyadmin you need to identify Prestashops database (left panel), it is the one you configured when installing PS. In that database you need to find specific tables. In my database all tables have prefix ps_ but in your case it may be something else so I will refer to it as xx_. So we need to find two tables, one is xx_module and the other is xx_hook. First you need to find out what is id_module of PayPal in xx_module. If you click on xx_module you will see a table of all modules with id_module assigned. In my case PayPal module has id_module 70. Then we go to xx_hook which is a table listing all hooks. We want to find out about paymentOptions hook so we look for it in the table. I found out that paymentOptions hook has id_hook 244.
Now the harder part.
Newbie friendly way:
We need to look at yet another table called xx_hook_module. This table connects modules to hooks. You should order the table by id_hook so that you can see all better how many modules are assigned to each hook. In my case I had to find id_hook 244 which had some modules assigned to it (Skrill was id_module 104). I copied the row with id_module 104 and changed id_module from 104 to 70 because paypal is number 70 in my case.
Alternative, SQL way is to just execute query:
INSERT INTO `xx_hook_module` (`id_module`, `id_shop`, `id_hook`, `position`) VALUES ('70', '1', '244', '1');
Values my be different in your case and remember to replace xx_. You put this query in SQL tab in phpmyadmin.