First, I've got to agree with you Thassias, the lack of codex creep is what makes specialist games attractive to me, too.
Second, I agree with horizon that Tyranids would make a good choice for a second fleet when you have a Tyranid 40k army. Assuming it's your cup of tea, having all of those parts laying around to scratchbuild a free custom fleet is immensely enjoyable.
Useability-wise I think they would rate as high as any other fleet. A fleet with a great number of damage and shield points overall and fairly straightforward weapons that complement each other well make for a fleet that is easily accessible for new players. It
is a little bit more labor-intensive to play them, especially for new players. Because their ships are not fully independent like other fleets, both in terms of their individual capabilities and also in the literal manner in which they must move, the special rules regarding Tyranid use take more getting used to than other fleets. They also tend to have a lot of models to push around, there won't be any turns where you just have to move six ships and go on to shooting.
Competitiveness-wise, they can struggle sometimes, and their worst opponent, aside from perhaps Necrons, is Eldar. Some of the main Tyranid fleet list's biggest weaknesses are that it is cumbersome to maneuver as a fleet and that it relies on a few key vessels for coordination. Eldar can take advantage of the first weakness to attack the second, and the lack of long-ranged Tyranid guns can make retaliation difficult. Just like any fleet, of course, you can build the Tyranid fleet to beat Eldar without too much trouble, or you can play a Vanguard fleet and simply swamp them with too many ships for them to possibly deal with. The trouble with the first is that you won't get as much of a chance to play around with some of the configurations that make Tyranids unique. The trouble with the second is that it can be boring for both players, although it can be fun with the right scenarios and mindset.
When it comes to their more general position in terms of metagaming, there are a lot of factors that keep them out of the top spot. If one is thinking purely in terms of competitiveness, Tyranids are definitely a min/max style fleet - a properly min/maxed Tyranid fleet can perform well against any fleet that is not designed to oppose it. The mega-hive list was quite powerful, the all-ordnance list can overrun most fleets, and the mass of escorts in a vanguard fleet is a thorny proposition that most fleets have trouble handling. The problem with this is the same as any min/max design though, it's boring! They can do well against all comers if they take large amounts of ordnance, but this playstyle is so frustrating to use that neither the player nor his opponent can usually stand to play an all-ordnance Tyranid list regularly. The Vanguard fleet has similar problems. It lacks the sheer tedium of an all-ordnance list (at least you don't have to pick up and put down 50+ ordnance markers each
player turn 
) but suffers from the same lack of player input. You just point your escorts at the enemy and charge them. If you take mostly Kraken you don't even have to decide whether or not to brace! The Mega-Hive list was slightly more entertaining, but with nothing but superheavy battleships for your opponent to avoid like a herd of elephants, there wasn't a whole lot to really do there either - plus, it was so reviled by the community that they entirely obliterated it with the FAQ.
Tyranids can be nicely competitive with balanced and interesting fleet lists as well, but, in-game, the slow interdependent way they play has always made them relatively vulnerable to nimble and powerful fleets, as well as fleets with powerful long-range weapons; they can struggle against Chaos, Eldar, and Necrons. One also has to be careful against other close-range fleets, such as Orks and Space Marines, because the superior short-range firepower of non-Tyranids (since much of Tyranids' offensive power is concentrated at point-blank distances) can make an error in timing critical. Due to the aforementioned mega-hive list, the new FAQ rules put rather harsh restraints on what Tyranids are allowed to field, both removing all choices for capital ships beyond light battleships and light escort cruisers, and creating a requirement for both a certain number and certain proportion of a particular escort type in every fleet. This makes Tyranids less attractive as a fleet before they hit the table, with relatively limited and much slower options.
Tyranids make nice balanced fleets in many ways though. The focus around a few key vessels can be as much an asset as a vulnerability, especially since their numerous ships allow them to soak up a lot of damage. They have no long-range shooting and little medium-range shooting, but their short ranged shooting options are perfectly serviceable, their unique weapons are quite powerful, and while their ordnance is slower than other fleets' it can be quite effective. (Unless you're playing against Necrons, lol) Even without focusing on ordnance they can create a fleet that easily defends against enemy bombers and torpedoes. Since Tyranids are more leadership independent than almost any other fleet they can be relied upon to perform special orders and do what you want of them most of the time. And the way Tyranids play means that watching their fleet at work can be much more entertaining than many others, like watching a snake swallow its prey or a school of feeding piranhas.
If you plan on using the Tyranids primarily for thematic battles against your Eldar then I would go with them without hesitation. (You said you're making an Iyanden fleet, right?
) If you want to also use your second fleet for more general battles and as a tool to introduce new players before they buy their own fleets, I would still consider Tyranids a good choice, but there might be other fleets that are a better fit. Like I said, their options are much more limited than other fleets, and their play styles aren't as varied. If you want a fleet with which you can do anything you want, aside from how you construct your models, they might be a better choice for a third fleet.
Of course, assuming you had the funds allocated for a second fleet
anyway, since you already have a Tyranid 40k army the only actual cost you'll have to bear for your 'nid fleet is time. If you're not short on
that you might as well get three fleets together...
