smiles and laughter won't make money enough to justify misuse of a great name.
For the project actually you could have used something like
MyHumorLinks.com or Links4Humor.com (both available).
I agree that MyHumorLinks.com or Links4Humor.com could have been perfectly good domains for this guys website.
But since he has links.com (a brilliant domain), I'd suggest that he use the sub-domain funny.links.com or humor.links.com
... and maybe later on he could branch out to sport.links.com, 3dtv.links.com and medical.links.com etc... The possibilities are endless.
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For the OP:
I think Links.com could work perfectly well if you expand your site and include keywords in your sub domains like I said above.
I think the home page could be improved. To me the home page layout kind of feels like something from a few years ago when the internet was still young and browsers didn't stick to standards very well so CSS and HTML were kept very simple. People might disagree but I don't think you need that black bar across the top on the home page. I'd move the About Us, Terms of Service and Contact Us links to the footer. I'd leave the submit a link and register links at the top, but they might look better if they were in tabs coming down from the top of the page (I hope you get what i mean). I'd probably run the digg and facebook etc links down the side of the page in a column because they look out of place. I do like that you just have that big image in the centre of the page - I think that works well, although at some point I'd consider hiring a graphic designer to redo the image.
On the link pages, again i'd get rid of the About Us, Terms of Service and Contact Us links. I'd move the 'click here for more links' graphic to the left - by the way I really like that graphiic... I feel like I want to click it

I'd have the home, register, submit a link and share links all aligned to the right of that black bar. I'd suggest changing the font or increasing the size of the current font because I find the current one a little hard to read.
Finally, i'd suggest adding some CSS styles to the form elements on the contact us, submit a link, and register pages - just make it look a little more up to date. On the contact us form, I wouldn't bother asking for addresses or anything like that. I know they are optional fields, but I don't think there is a need to ask for that information, and I think it could possibly turn people off completing any of the form at all. I think that with a form, the less fields that are present the better. If people have a reason to give you their address, they will be able to provide it in their message for you.
EDIT: I noticed some of the material in the links could be offensive or not suitable for some people/minors (i.e. humor of a sexual nature). I had a look at the terms of service and noticed that you say that users must be over 18 to use your site. Thats perfectly well, but you should probably make it more noticeable (like a big disclaimer on the homepage). Like most people I click first, think later. I don't think I have ever read a site's terms of service before I used it - and I think most people would be the same.