What reasons are there to mine and support Litecoin?
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.