I've got a different question than I've read about in other threads. I know the standard rule is to arbitrate guys twice, and then sign them to a long term deal (assuming you want to keep them around long term). My question is...if I sign a guy to a long term deal after his 1st arbitration, or maybe before arbitrating him at all...will it increase the odds that he wants to stick around and sign another long term deal after the first one is over?

Nope. That's determined almost entirely by his Patience rating, far as I know.

Ok, thanks damag

A few seasons in and have missed a couple of long term re-signings as I went to arbitration a 3rd time. Is there a recommended way to monitor or specific time in the yearly cycle to offer long term before third arbitration?

Every season there is a free agency period you can offer any of your free agents a long term deal then. So if they have been arb'd twice prior and you want to keep them do it then. You can do it during the arb process if you like. There is a link for offering a long term deal in that page.

To be specific - After budgets are set, the Arbitration screen shows Long-Term in each player's negotiation link.
You have that one day to use that link for a LT deal, so you should have already decided when setting your budget.
The next day is Arb day. The link changes to Arbitrate. However when you go to the individual arbitration there's a link in the middle of the page that is your last chance to offer a LT extension instead of Arb.