My main argument for correct spelling online is that we either have to spell correctly or build search engines that can understand your gibberish. If you can't spell, you can't search, you can't find information and you are left far worse off than those who can spell. The old standby argument of "you can still understand me, so what's the point?" won't fly with a search engine.
If your spelling is merely bad, search engines can cope. If your spelling is awful, Google will choke on it and you just don't get your results. I have tried making this argument to people who seemed to simply not care at all. The exact response was "and yet I find myself unmoved".
Well, you should be. Bad spelling puts your information further away from you, whether it's you doing the search or you publishing the text. If you can't find what you're looking for and your search engine can't make sense of what you're saying, you don't get your information. Now look at it the other way. What if someone tries to publish an important medical study, but misspells a keyword. That study might be critical to your work, but you'll never find it if you search correctly.
I may be overstating the case a bit, but if the English language degenerates to a state where nobody even spells anything correctly any more, the benefits of the global communications network we have striven to build start to dissolve.
Mokalus of Borg
PS - Spelling for names is particularly important.
PPS - Compensating for bad spelling wastes energy.