To be honest, I don't believe that option is available in the standard browsers for phones. You can try searching for additional browser apps online to see if any have that option, but I don't see it being a common feature at all. Plus you also risk getting apps that have ads or in app purchases, if not worse.
You can easily rename files once on the device. You should have a file manager app and can go through that to rename them. Or, if by some chance there isn't that kind of app on the device, you can always download one.