Just took over a new team, and this player is asking for $8.4 million in arbitration (his final arb year):

http://whatifsports.com/HBD/Pages/Popups/PlayerProfile.aspx?pid=4486695

For background, this team went 51-111 under the previous owner, and honestly, I'm not sure how they even won that many. I won't contend this season.

My question is, would it make more sense if I cut him loose and save the money or offer him arbitration and let him leave as a (hopefully Type A) free agent next season? I don't want to sign him long term, since he's expensive and not a very good hitter. But I'm not sure if the extra draft picks are worth the expense. Any thoughts?

If he isnt very good, and a type A, he might not get signed next season if you arb him. Then you will have spent the money this season for nothing.

True, he's only a .250 hitter, but he's already won a gold glove at 2b and steals 30-40 bases a year..  He also scores a lot more runs then you'd expect from his OBP, probably because of his speed. Someone will want him.

If they don't, you can bring him back on the cheap after FA is over. 

Looks like you have the cap space to carry him for one more year. Arb him, shop him and see what you can get in trade while he's still under control. (If there's a market for him, he might cost another team a lot more in free agency.) If you can't get a good return, let him walk and take the comp pick or picks. Mike is right, if no one wants to pay the comp next season you bring him back at a more appropriate salary.

Agree with the consensus.  I actually like him as a player-- on a team that runs a lot he's a championship-quality 2B/CF, especially if you have a backup that hits LHP well, so you can bench him 20 times a year against LHP with a low VsL-VsR.  If you think you're only a year or two away you might even long-term him instead of arb'ing him.  Otherwise he should have significant trade value.

Despite what I said about this player, don't overrate Type A compensation in general.  It's often a low-ish supplementary pick (around 40) and a low-ish second rounder (around 80), and that's usually a just good bench player and a marginal bench player.