When a player becomes a free agent, their current team retains "Bird Rights" that translate into an inflated cap hold that counts against the team's salary cap. "Bird Rights" were named after Larry Bird as a way for teams to keep their players despite being over the salary cap. The cap hold was added to make it so teams can't sign new players and THEN re-sign their existing ones. You can't have your cake and eat it too.
In general, "Bird Rights" give the advantage to the player's current team. A team with "Bird Rights" on a player can offer one more year (5 vs. 4) and higher raises (7.5% vs. 4.5%) than any other NBA team. In the new CBA, even sign-and-trades must now be at the lower numbers rather than the higher. Only 4 years with 4.5% raises on sign-and-trades now.
In Steve Nash's case, "Bird Rights" are likely a non-issue. Nash's last contract, when he was a young 35, went for 3 years with a DECLINING salary. Going forward, he is asking for a 3-year deal and, with presumably diminishing skills, there is no reason to expect Nash's next contract to include raises either.
Under those parameters, "Bird Rights" mean nothing to the Suns or Nash.
The only other value that "Bird Rights" offers is for teams to exceed the cap to re-sign their own player. However, you can't both sign new players AND keep your existing ones if the combination puts you over the $58 million cap.
The Suns currently have $31 million in guaranteed salaries for the 2012-13 season (Gortat, Frye, Warrick, Childress, Dudley, Morris, and 1/2 of Telfair's deal), leaving a little more than $26 million to sign new players or re-sign current ones.
Factor in the logical move to retain "Bird Rights" on younger players Robin Lopez, Shannon Brown and Aaron Brooks ($16.9 million between them) for negotiating and trade leverage purposes during free agency, and you're down to only $9 million in cap space on day 1 of free agency. Sure, those guys will eventually sign for less somewhere. But until they do, that's the collective hold on the Suns cap.
Nash's cap hold is another $17 million. Grant Hill's is another $9 million. If the Suns keep those guys' "Bird Rights" as well, suddenly there is NO MONEY to spend in free agency beyond the mid-level exception. Ugh.
Renouncing Nash does not diminish the Suns' ability to re-sign him. He won't be asking for anything special that "Bird Rights" offers. Same with Grant Hill.
This is why you are very likely to see 2 major things happen in June:
1) Suns renounce "Bird Rights" on Steve Nash and Grant Hill, creating a wave of new 'Nash is GONE!' articles
2) Since the first move only gives them $9 million to spend, which is lower they likely want, Suns also use amnesty on Josh Childress' $6 million cap number
With those moves, the Suns starting number of spendable money on day 1 of free agency is $15 million.
Without those moves, the Suns have NOTHING to spend.
Nash has said that he wants major improvement to the talent on the roster before he considers re-signing. Without the starter bucket of $15 million, the Suns can't honor those wishes. They can later re-sign Nash and Hill with the freedom created when Lopez, Brown and Brooks sign (with anyone) for a lot less than their collective $16.9 million in cap holds.