What reasons are there to mine and support Litecoin? There are many examples of Bitcoin being accepted as a means of exchange in online transactions.
I haven't found any such examples for Litecoin.  

So, what is the value in LTC mining? What is LTC's purpose if you can't buy anything with it? Do people believe that LTC is in the process of becoming a currency that will one day be accepted in the same online transactions as Bitcoin?
This is simply my opinion on why I get into altcoins. I like to diversify my portfolio. In my eyes any one currency could overthrow another currency. 

What I usually do is mine a decent currency, then use that currency to invest in certain things. For example and this is simply an example and I'm not using it to spam. I would mine litecoin then spend my litecoins on hashing on CEX or BleuShares or something else. This starts to make me stronger and gives me more. 

I also sometimes buy metals such as gold and silver with my crypto currencies. But I'm a firm believer of diversification. 

There was a twilight zone episode where theives steal gold and sleep for 100 years to find out gold can be manufactured in the future and the gold was worthless. I don't see such things happenning anytime soon, but it is nice to not put all your golden nuggets in the same basket.