LOL (laughing out loud)
An undated online post by Wayne Pearson claims LOL was first used on a BBS chatroom called Viewline in Canada. He says the usage then spread to the GEnie service. As noted earlier, GEnie was a competitor to CompuServe, Prodigy, Dialcom, Dialog, Delphi, Lexis/Nexis and other proprietary networks back in the day. Pearson explained that other variations -- such as *smile* -- just didn't convey enough emotion. At the very least, Pearson's post shows up first in a search for "origin of LOL" at Google.com.
Zimmer notes that the first usage of LOL appeared on the aforementioned "FidoNews" newsletter in 1989.
TTYL (talk to you later)
This one seems to be of a more recent vintage, since it doesn't appear on the Jargon File. Once again, it likely became popular on AOL where chatting is more immediate and one to one. (There's no reason to say TTYL on Usenet, for example.)
However, Zimmer points out that TTYL appeared as early as 1985, according to textfiles.com.
ROFLOL/ROTFL/ROFL/ROTF (variations of the concept of "rolling on the floor laughing out loud)
ROFLOL was used by someone named Dave Alexander in a Usenet post to the group alt.rock-n-roll in 1992.
ROTF (without the L for "laughing") appears in the 1990 Jargon File. The next year, the Jargon File contained the term ROTFL.
ROFL (without the T for "the") was used in a Usenet post to rec.ham-radio in 1989, so we know it has been around for at least 19 years.
RTFM (read the *blank* manual)
Again, likely invented in the '80s on Usenet, where scolding people who ask inane questions is commonplace. Also, thick manuals were once included with software back in those days and reading them was even considered a good idea. The very first post I could find on Usenet that includes RTFM was in 1983 and refers to the term having been used by "VMS people." VMS is an old server operating system.