|I told this list a horrible lie a few week ago in a conversation about Webmin. A bit OT, but I apologize if some people were misled...|
I said Webmin was slow on the Raspberry Pi. That's not true. *Iceweasel's* slow on the RPi. Webmin is fine on fine on Raspian's default browser. Unfortunately, that browser doesn't do frames, and Webmin does, so half of Webmin is missing.
Most of the lightweight browsers don't, I found out. Except for Evacuation or whatever Gnome is calling it this week. Web? Evince? Evacuation's quite a bit faster than Iceweasel on an RPi3, but installing it brings in an awful lot of Gnome bling. I'm on XFCE to get rid of that stuff.
But there's a browser that does frames, starts up quickly, is real fast, and is relatively lightweight: Luakit. It's kind of strange if you aren't a Lua programmer -- a lot of configuring is done by futzing with Lua code files in /etc, and a lot of its commands are CLI from Vim. It feels like a lynx that's been dragged kicking and screaming into the GUI world so it can display pictures.
It's supposed to be for people (from the home page) "that have a too much time on their hands." But it does Webmin 1.78 magnificently right out of the box, with Jessie Raspian.
I'm using it now just for Webmin, and Iceweasel for pretty things, until I get used to Luakit. It's very cool, but it doesn't have Iceweasel's plug-ins, like adblocker, out of the box (I had no idea what's happened to the web in the last 15 years). That's not a problem running Webmin, and there's likely something in all that code in /etc that will do it, but I don't know enough Lua yet...
Anyway. I think Luakit is what you're looking for to run Webmin on the RPi (I haven't tested it for purity at w3c, but it's good enough for Webmin). Gnome's browser will also work, but it brings with it more garbageBits than I care for. None of the other RPi-worthy browsers I looked at will do at all.