How can I update my jailbroken iPhone 3GS to iOS 6.1?
Your best bet is to use iTunes to update it:
(WARNING: this will factory reset your phone. You will need to ensure all your data is appropriately backed up - you'll also have to re-jailbreak afterwards.)  

Put the phone in recovery mode and plug it into the computer.

Next, open iTunes. It will tell you it has detected an iPhone in recovery mode and give you the option to update and restore it. Do this. (It will download about 700 MB of data and then factory reset the phone.)

Boot your phone up and it will be running iOS 6.1.6! You'll need to re-jailbreak.
An iOS 6.1.{3+} jailbreak needs a tool called p0sixspwn to be untethered - be sure not to forget that! (I won't go into detail here, but there is plenty of info online about this extremely simple step.)

Enjoy!