The best option would be to tweak the low memory killer's minfree parameters.
The Low Memory Killer is the mainstay of Android memory management. It's a more elegant approach than the Linux oomkiller and it works proactively to maintain the free pool rather than only kicking in when you're completely out of free memory. It separates apps into several categories for killing if the free memory pool gets below certain points. They're generally in the following order, from first-to-die to last:
EMPTY_APP - These are apps that are not doing anything, nor waiting to do anything. They're just sitting around in memory.
CONTENT_PROVIDER - These are background apps that content to active apps (e.g. The play store uses one to check for updates periodically. HTC Facebook sync is another common example).
HIDDEN_APP - These are sitting in the background, not doing anything, but are still alive and possibly waiting for something.
SECONDARY_SERVER - A server running in the background to provide services for a currently running app.
VISIBLE_APP - This is an application that is in the background, but it is presently doing something.
FOREGROUND_APP - This is what is presently running and on-screen.
If the free memory pool drops below a certain amount (e.g. 80MB is the default on my GS3), the system will first start killing off anything listed as an empty app until the pool is back above that line. If after it kills off all the empty apps the memory is still below the next line (e.g. 64MB), it'll start on the content providers, and so on, until eventually if only the foreground app is taking up all the memory (On my GS3, if everything except the foreground app has been killed and there's still less than 32MB of memory free) and threatening the system, it'll get killed off eventually.
Coming back to your actual question, what we want to do is adjust these values down, so the killer will kick in later and hopefully not kill the browser when you're still wanting it open.
The app MinFreeManager will allow you to adjust these values. Alternatively, they can be edited directly in
/sys/module/lowmemorykiller/parameters/minfree where the parameters are in pages (4 kilobytes, so a value of 8192 means 32MB as ((8192*4)/1024=32MB), and listed in reverse order from what I listed above. Both of these will require root. If you don't have root, there's basically nothing we can do to help.
In your case, the HIDDEN_APP parameter (4th item in the minfree file) is probably what we need to change. For example, this parameter is by default 56MB on my GS3. Cutting this in half to 28M or using the mild preset in MinFreeManager would be a good starting point for tweaking.