Browser options for older iMac

Firefox would be your best choice if your dad is still using Snow Leopard. You might want to advise him not to use his iMac for any kind of financial browsing such as accessing his bank, etc. Apple no longer provides security updates for Snow Leopard. BTW, if you mention which iMac your dad owns (year and model) we might be able to tell you if he can update to a later version of OS X.

 

If it is a 2007 iMac then you can go up to 10.11.6 (El Capitan).

El Capitan is also available from Apple as a free download since you are upgrading from Snow Leopard and can not install Sierra:
https://support.apple.com/en-us/HT206886
It may operate a little slower than you like. If that doesn't suit you then you could try Mountain Lion (10.8.5), which can still be purchased from Apple. Let me know if you need the link.

Make sure you have a good backup before installing in case it doesn't go well or you don't like it and you want to revert. I would recommend making a clone backup. This is important!

 

Sorry I can't answer. I left 10.7 in 2012 and no longer have a system with 10.7 to test for you.

EDIT:
Here are some threads in the Lion forum that might help:
https://forums.macrumors.com/threads/best-web-browser-for-10-7.2018761/#post-24009134
https://forums.macrumors.com/threads/10-7-5-lion.1992557/

 

If it's a 2006 iMac Core 2 Duo, then Lion is as high as he can go. And I believe that version of Safari will not work for secure browsing. Best bet is to stick with a Firefox version which does provide for a measure of security.

 

The main problem with older browsers is that they don't support TLS v1.2. You need Safari v7 or later (macOS 10.9+). However, there are other browsers that support TLS v1.2. On Snow Leopard you can try: Opera 12.16+ (also Opera 26), Firefox v27+, Chrome v30+. Here is a full list of browsers.

On Leopard (OS X 10.5) the only browser that supports TLS v1.2 is Opera 12.16. Some ciphers are missing, but it should work with most sites. Unfortunately, in 2016 they updated Opera 12 only for Windows, not for Mac.

You can test your browser here:
https://www.ssllabs.com/ssltest/viewMyClient.html

So in addition to Firefox and Chrome, you may try Opera 12.16 or Chromium based Opera 26 available at:
ftp://ftp.opera.com/pub/opera/ (open "desktop" for Chromium based versions or "mac" folder for 12.16 Presto based versions)

 

Didn't check the TLS v.1.2 specs of these (edit: Firefox ESR 45 supports TLS v.1.2), but on 10.8. Mountain Lion I run Mozilla Firefox Extended Support Release (ESR) 45.x that will still work and still gets security fixes on 10.6, too. On the release plan I can see at least support until June 2017, though this timeframe for the last release could be extended or not. Some time ago it was announced last release for January 2017 and it was luckily extended!

Info on https://www.mozilla.org/en-US/firefox/organizations/faq/
Download the ESR 45 release on https://www.mozilla.org/en-US/firefox/organizations/all/

Then I've found, but didn't test:
NetSurf
Roccat

IMHO, Safari is such an old release, that it can't be really considered as a secure browsing option, even on 10.8. Personally, I think that 10.6 is a great OS and I wouldn't update to 10.7 as I can't see a real advantage in doing that.
Except you absolutely want to run the iCab Browser on it.

Another long term option is to find some actively maintained open source browser that will compile on 10.6 and compile it by yourself. Some starting point could be at https://en.wikipedia.org/wiki/List_of_web_browsers

As there are projects like Classilla for Classic Mac OS 9 and TenFourFox for Mac OS X on PPC that are both based on Firefox ESR 45 code, there is the hope that ESR 45 will be maintained for a longer period of time or that there will be other projects arising that will somehow maintain browsers for Mac OS X 10.6 and lower.