Hello, I'm new here. @Thanatermesis asked me to suggest some browsers for the Stable version of Elive and then suggested I bump this topic and share some suggestions.
Please note that I haven't tried the Stable version of Elive or Elive itself in more than 10 years but I'm eager to give it a try on my second computer so I can't guarantee that the options will work. Also, I find it difficult to suggest anything specific unless we know what services, extensions and websites one would access on a daily basis. Some of them will not work on anything other than a browser based on Chrome or Firefox and, in that case, there isn't much to choose from. 
But, perhaps, you'll find something that will fit your use cases below. 
Since I wasn't sure if there was any justification in getting new builds of Chromium to compile on Elive based on Wheezy, I didn't mention it and thought about the lightweight options that might be easier to build on a less modern system. From the top of my head, Otter Browser, Falkon, qutebrowser could be useful to some users, though the latter seems to be moving to QtWebEngine and doesn't recommend to rely on QtWebKit anymore.
Midori used to be a very lightweight backup browser, but it's now being developed by some foundation that moved it to Electron so it's not as lightweight as it used to be. If Elive can use the version in Debian Stretch, it might be useful to some.
Maybe something very minimalistic like Badwolf could work as well?
I would add links to all of them, however, it seems I don't currently have permission to do so.
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