Ah Piracy..
We can split this into a couple groups (I know how much folks love to be grouped, but work with me here).
1. Corporate Raiders - these are "officially sanctioned" pirates, in the employ of a given faction or corporate interest, who specialize in raiding their competition. They're getting paid to rob, plus bonuses for anything else they might happen to come across. We already have "Liberate" missions that turn up on the mission boards to feed this particular form of Piracy, but this can be vastly refined.
2. Freelance Pirates - Thugs, thieves, and even your "gentleman pirate", working for no one but themselves. Once the scourge of the high seas, now the bane of the black. Like the Corporate Raider above, this too is in need of some major refinement.
So, how to begin to refine this? It ultimately comes down to the mission system, and to the ranking system. It also comes down to systems in general.
As it stands right now, there are no particular havens for pirates of any kind - no den of thieves for them to gather, and little in the realms of information for them to gleam.
--and here's the edit--
To make Piracy actually viable, Pirates are going to need stations in Anarchy space. I would expect to see these take the form of Asteroid bases (something we've been told is in the works), or planetary installations, as these would be a bit easier to keep the whereabouts of obscured from Johnny Law and more importantly, from Bounty Hunters. These may not be as feature-rich as more "legitimate" stations. I wouldn't expect much in the lines of commodities at these kinds of places, nor Contacts for claiming bounties or combat bonds. But a vibrant Black Market, paying more than the standard rates - more along the lines of the more legitimate Commodity Markets.
I also wouldn't expect much to be offered in Shipyards, but Outfitting I would expect to be well supplied. Clearly a lot of effort will be involved to make these wretched hives of scum and villainy both viable and useful.
Mission boards here would be dominated by missions suited to pirates for those Corporate Raiders looking to ply their particular trade for shady corporate officers looking to put the hurt on their competition though such means.
Another unique feature of these sorts of stations would the Rumor board. Much like Tip Off missions, the Rumor board would provide locations of valuable shipments, lost ships full of precious cargo, and the like - suitable for the freelance pirates to sate themselves on ill-gotten gains. These missions would require a bit different handling - no cash payouts, but certainly rich with Reputation.
And for the Pirate, Reputation is what the "game" is all about. Whereas the working stiff builds reputation and influence with various factions, the Pirate's Reputation is all their own - reknown (or infamy) for their exploits. This serves the same role as Pilot's Federation ranking, indicating the overall skill of the pirate. Similar to a Trader's rank, it is based largely on the value of goods stolen and successful sold. Factors that affect the value of a sale, such as being scanned with a hold full of stolen goods would reduce the amount of Reputation gained.
Destroying other ships will also not aid one's rise through the ranks, as this does not produce cargo to resell.
Of course, this is only the PvE side of the coin. PvP Piracy will also benefit from these dens, as places to pawn their goods in relative peace, and to supplement their incomes, but they will also gain the benefits of Rank as well. I've not yet fully thought through what those benefits are, and am open to suggestions, though I think these should be on par with current benefits - access to otherwise locked system, the opportunity to obtain certain rank-locked ships, and perhaps some influence when it comes to dealing with PvE pirates - making lower ranked NPC's more apt to submit, to eject cargo, and to eject cargo of higher values, rather than facing the wrath of a greatly feared pirate.