I don't understand the "no entry sign" but if 0 is truly safari and it is in the root of your boot directory below is a terminal command to move it back into /Applications renamed as Safari.app.
Open Terminal (in Utilities) and copy/paste the following line into the terminal window:
sudo mv /0 /Applications/Safari.app
Hit return. You will be prompted for your admin password. It will not be echoed as you type. Hit return again and Safari.app should now be in /Applications.
The icon at the top level - currently named 0 - is the Safari icon and double clicking it does indeen launch Safari so I'm assuming all is well. The icon under the no entry sign is greyed out (as is the no entry sign itself) is also Safari and again, double clicking that launches Safari. Is there a possibility that one is acting as an alias to the other? (There is no alias type arrow on either.)
Still in terminal, type ls -ld followed by a space. Then drag the greyed out icon into the terminal window. You should have a line looking something like:
ls -ld /gray-out-filename
Hit return and post results. You didn't originally say there were two files. I thought it was only that /0 file. As I said earlier I don't know what that "no entry" symbol means in this context. But it certainly isn't an alias.
Sorry if I didn't make it clear - I said "moving it back to Applications seems to create a copy (which does allow me to rename it) leaving the original at the top level." What I meant was, I tried to drag it from the top level to Applications. I expected this to do a normal move operation but it didn't, it created a copy just as I would have expected if I'd dragged while holding down the Option key. So the original remains at top level, the copy in Applications is greyed out (but seems to work).
I expect I can delete the copy in Applications and try your original suggestion to move using Terminal but I'm not seeing my friend for a few days - I'll post back what happens.
Thanks again for your help.
Delete it. Use another browser like Firefox to go here; you can download Safari and install it again.
Prevention tips: NEVER let little kids use your user account. Give them their own non-admin user account to play around in, from where they cannot mess anything up.
Sorry if I didn't make it clear - I said "moving it back to Applications seems to create a copy (which does allow me to rename it) leaving the original at the top level." What I meant was, I tried to drag it from the top level to Applications.
I think I understood what you said, i.e., you had two copies in your root directory, one called "0" and one grayed out with what you call a "no entry" sign on it. You said both seem to be valid versions of safari. So I gave you the mv command to move "0" into /Applications and rename it as "Safari.app". I also asked you to do a ls about the second one with the "no entry" sign on it just to see the ownership and permissions.
Are you claiming the mv didn't work?
Did you do the ls? It wasn't going to remove anything. It was just informational.