I've played with Tiny before... and its not exactly for the newer to Linux crowd. We put it on a 33Mhz 386 8Meg RAM. It ran fine, but was a beast to configure. You might just want to go for an older slackware
that actually came on a pile of 3.5s. All of these older versions are archived at the Slack site. 3.x or 4.x should be alright, just as long as you get something in the 2.x kernel family.
16 Meg of ram is going to be too little for either an ftp or nfs mount to a remote site with a newer distro, so dialing in to do a remote install is probably out.
Either way, good luck with the old machine.