I think I'm going to preface this with the statement that application does matter considerably. To help illustrate this, I'd like to point to
an excellent review by Hardware Secrets showing the performance of mayonnaise when used as thermal paste.
It's not always the thermal conductivity that counts, it's what you do with it.
And what's better than copper? Silver?
If yes, why we do not see any heat sink made of silver? Otherwise disregard this post.
High cost, very small market (if any), not a whole lot better than copper.
Here's a list of common materials. measured in W/(m K), followed by their price per pound. Higher numbers indicate better thermal conductivity.
Aluminum: 205, $0.83
Copper: 401, $2.87
Silver: 429, $252.64
Gold: 310, $17,357.42
With copper, we pay a little over three times the price of aluminum to gain about two times the performance, with silver, we'd end up paying 88 times the price for ~7% more performance.
If I'm not mistaken, gold is expensive for its electrical conductivity and resistance to corrosion. As an unintended side effect, computer cases would need to be
very secure.
copper should transfer heat better then nickle
Indeed. Nickel only scores a 91 on the above chart. I think it's, again, used for resistance to corrosion.
inb4 someone asks for diamond heatsink
This made me chuckle because I was thinking the same thing.
Diamond is a very,
very good thermal conductor. Time to continue the chart.
Diamond: 1000 $???.??
Pricing gets very hard here since the price of a diamond increases exponentially for its size, but an industrial grade diamond (one unfit for use as a gem, and as cheap as it gets), is about $12 a karat. One karat is about 0.007 ounces. That puts us at $1,714.29 an ounce. 16 ounces in a pound gives us a total of $27,428.57 for one pound of the cheapest diamonds one can source today.
I'd be pretty confident to guess that the process to manufacturer a bunch of random diamonds in to something that can be used as a heatsink is also very, very expensive, further adding to the unlikeliness of this ever happening.
Moral of the story, copper isn't cost efficient, but the cost is low enough for enthusiasts and applications where the increased performance is critical. Aluminum is good enough for average jobs.
I like analogies and I can't get this one out of my head, you can get a hamburger for $1 and it'll get the job done, but most people will spring for the $2 cheeseburger.
Sources: Google, mainly
engineeringtoolbox.com for thermal conductivity,
vincentmetals.com for metal prices, various Googling for diamond pricing.
Disclaimer: Prices are likely not accurate, given the volatility of the market and my laziness to find consistent and/or up-to-date sources. It should be a pretty good ballpark though.
....i hope you enjoyed my bad quality photos! 
