I like reddit and have made it part of my daily internet schedule. I had to give it up for a while because it was too much of a time sink but then I discovered the Reddit Enhancement Suite which allows you to hide a set of subreddits from r/all, and which makes the thing much more manageable.
There are lots of things I would like to do differently, though, if I was writing a reddit-like website. One thing I don't really like is the subreddit system - rather than one big website it's more like thousands of little websites (of various sizes) which you can combine into a single feed. But it's hard for somebody submitting a link to know which sub-reddit(s) to submit it to - if you submit it to a large one it'll probably get lost in the noise, if you submit it to a small one very few people will see it, and if you submit it to too many you might get accused of spamming (not to mention having your upvotes split between submissions). Then there are massive flamewars about whether a particular post is appropriate for a particular subreddit. Readers have a similar problem - which subreddits should you subscribe to to see all the news you want to see without having to wade through too much rubbish and duplicate posts?
I'd rather have a system of tags instead of categories - people can apply multiple categories to a single post and you could set up complex rules about what you want to see ("I want to see all posts in category X unless they're also in category Y and aren't made by user Z").
Another problem is "how do I get things that I've made that I'm proud of onto reddit?" Being an honest user, I want to avoid submitting my blog posts myself ("It's not strictly forbidden to submit a link to a site that you own or otherwise benefit from in some way, but you should sort of consider yourself on thin ice.") except for things like r/somethingimade. I had a "submit this site to reddit" button here for a while but to my knowledge nobody ever pressed it. I want to contribute to reddit and make it better for people who like to read the sort of things I like to read (which are also the sort of things I like to write), but since most of the things I read on the internet I find there and most of the others have already been submitted there, there is little opportunity for that (meanwhile, the large chunks of the site are flooded with useless memes and duplicates). So I'd like a reddit-killer that allows and encourages people to submit their own stuff - maybe even have the ability to submit an RSS feed so that new entries on a blog are submitted automatically.
That would mean a lot more posts in general, which means you'd need a more scalable way than reddit's "new" queue to allow people to find things that they're interested in. I envisage a "people with similar tastes to you also liked" system a la Amazon or Netflix (though Amazon's recommendation system is almost completely useless - no, the fact that I bought a bicycle wheel doesn't mean I'm interested in buying lots of slightly different kinds of bicycle wheel). So I'd want to automatically "like" or "upvote" my own posts, then they'll be shown to people who have liked my posts in the past, they will then upvote causing it to be shown to people similar to them and so on until it's traversed the internet or been downvoted into oblivion by the users most similar to me. Comments made in reply to posts or other comments could be handled the same way - I'd see interesting comments on my feed interspersed with the posts rather than having to go and check interesting posts/comments for interesting comments individually.
Yet another difference I'd like is to do with timeliness. Reddit is all about the newest, hottest thing that's just happened - posts more than a day or two old don't tend to stick around (unless you visit a quiet subreddit specifically). But I'm not not interested in the best new thing, I'm just interested in the best new thing I haven't seen yet. I'd rather see an old-but-good post I haven't yet seen than the best of the new stuff if the older one is better still. Now, this does have a danger of making the site an even bigger time-sink than Reddit is (since the pool of all stuff ever is bigger than the pool of stuff that's new in the last couple of days), so there would need to be some way to make it managable. Once you start downvoting a certain percentage of things it shows you, that probably means its time to stop showing you stuff. Also, the ability to say "don't show me anything new between the hours of 9am and 5pm Monday to Friday" (for example) would probably be a useful feature.
If you know your friend is on this site and has an interest in a particular sort of post, you might want to make sure he sees a particular post by recommending it to him - then the site will boost the score of that link for that friend (to an extent determined by his measure of your reputation). If the site becomes sufficiently popular then sharing links this way is far superior to emailing them to your friends or posting them on your facebook page, for both recommender and recommendee. With text posts and privacy controls, it could supplant these things altogether - i.e. you could read your email through the site and it would rank the unread items in your inbox alongside other things recommended for you. In other words, it becomes your "what have you got to show me" interface to the web just as Google is your "I'd like to search for things about" interface and Wikipedia is your "I'd like to find out the basic facts about" interface.
A particularly time-consuming part of reddit is watching videos and listening to audio, but these do have the advantage that they can be done in the background. Perhaps this site could take advantage of that by having a video/audio feed separately from the text feed, which plays in the background while you're reading the text feed and doing other things.