Mike is correct that it might not necessarily be the first arb year for the player. If, for example, a player with options remaining and 3.000 years ML experience is signed via arbitration, then demoted and spends the whole year in AAA, he will still have only 3.000 years ML experience and be up for arbitration the following year as well. See
Roger Balboa as a similar example. He'll still demand arb. 2 more times, but his demands this time are low (in line with someone who is having his first arb. hearing).
$ - The dollar amount listed represents the player's arbitration demands for the selected season. You may offer the player any amount of money you deem reasonable during hearings and an arbitrator will rule immediately.
(This is from the HELP, which can be found towards the upper-right of HBD pages).
When you said "I gather that it makes no sense to offer arb in the first year of eligibility? Correct?" did you mean you gather is makes no sense to offer a LT deal in the first year? If so, that is in line with the conventional forum wisdom. There may be certain circumstances where it is worth it, in my opinion, but most often it is not.