One I know could handle it. Provided you can get it to install ( such as using nunol's adding a swap before installing tech. At least he's the 1st person I saw mentioning that approach on the boards.) Might even install w/o having to do that. Cause it's specifically intended for low resource systems. Vector linux light 6.0, on my comp at boot up and idle uses 43mbs/ram.
It'd be the one I'd go with. My comp has 512mbs/ram. Which is considered low by today's standards. So why I've gotten so interested in lighter weight distro's. AntiX 11, could probably also do it (65-70mbs/ram at idle on my box). Though with any of the popular full browsers running. Think it'd be furiously using the swap and laggy on a box with 128/mbs ram.( antix, might even flop out with a brower running.) VL 6.0 and Antix are both full featured linuxy goodness.
Another that's supposed to be really good on system resources while still being really nice. CTKarch, haven't tried that one myself though. Don't feel like adding another distro atm. But seen things to suggest it could maybe be even lighter than VL light, shrugs.( edit: With a tad of tweaking anyway). Which would be friggin amazing if it is. But don't quote me on that. Know it's supposed to be super light regardless.
One thing you're up against n no matter which distro you want to go with. You might consider using a low resource web browser. Midori was the best I found for that. Though midori in earlier versions also found to be buggy. That may've been sorted out in the newer versions recently released for it. Don't know for sure about that though. With 128mbs/ram, thinking use of a web browser is where you're likely to get sunk otherwise.
You might have to get good at logging out n shutting off your GUI, when you do web browsing. Then learn to fire up your web browser in CLI ( command line ) Restart your gui, when you're done. Pain in the butt, with 128mbs going to have to get a bit crafty if you want a good computing experience on it.
Summin else that occurs that could really improve a persons experience with a low resource box. Adjust your OS's swappiness value. Just google vm.swappiness=10 to get the general idea. I mean once you get one you like installed. With the default value of 60 which seems to be the standard most distro's come set at. On a 128mbs/ram system, it'd be constantly using the swap and doing disk writes at that value setting. No need for that if there's still available RAM free. Harder on hardware n has to make things more sluggish. Swap shouldn't kick in until you've topped out the boxes RAM n it's actually needed.
Long story short ... I'd say look at Vector Linux light and CTKarch. Won't really know until ya try. Wouldn't be surprised if the Magic Mint team have releases that could be made to work on it. Oddly haven't really tried Mint's low resource stuff. So can't offer opinions on them atm. Guessing there would be mucho tweaking and trimming + other obstacles you'd have to sort out if your heart is set on summin Minty with only 128mbs/ram though.
Note: To Mint HQ, might not be a bad idea for you gals/guys to apply your dev talents to coming out with some versions for old, neglected hardware. Might help improve Mint's adoption. Are plenty of people in the world with oldish hardware ... n low system specs. Just a obs ... nobody makes linux as Minty as the people behind Linux Mint.