That's a mighty good question....Lil I rarely ask the gods "help". I'm rarely brave enough too. You never know what you are going to get from them...especially Odin.
Interesting question. How "big" was this thing Odin gave you? That's one way to decide. Then ask yourself what would Odin like? I mean 3D glasses and tickets to 3d IMAX movies are useless, as he doesn't have depth perception.
Usually a bottle of mead poured out as symbolic offering. I've pledged to make a working tool for someone who couldn't afford one if something came about. It did and I made the tool.
Mostly it's gift for gift - I acknowledge and give him honor, he occasionally takes note of me, guides (or smacks me upside the head), or tests me.
Added: Eiliat - if you aren't Asatru, why are you answering? Do you know anything about Asatru?
Am not Asatru, but I am a Norse Mythology nut. All I know is that Odin was sort of a fickle bastard, first giving his assistance then turning the tables on you. So if you get help from him, watch out; you may be on his bad side tomorrow.
I heard he really wants a back scratch er. Buy a good quality one and burn it on an altar. If it was a big favor, then get him an MP3 player and headphones.
You would thank him with a sacrifice -- usually a libation such as mead or beer, a votive offering of amber, or incense. I believe the offering should be given when the request is made, and then again when the request is granted, in thanks. They say, "a gift for a gift," but I personally see it more as building a relationship with the Gods, rather than simply "give to get."
This is not a concept unique to Asatru. It is also found in Hellenismos.
"The glue that ties this all together is the concept of khakis, the reciprocity so implicit in the practice of Hellenismos. The author discusses the application of khakis to everything from formal group ritual to intimate personal interaction with the gods. From the simplest devotional act, to prayer, to divination, to mysticism, the principle of reciprocal favor governs the heart of this religion and lets each worship per encounter the gods on a real and profound level."
The one time I ever asked Odin for something, I gave blood (read: little tiny scratch blotted up on paper and buried).
Personally though, I'm not much for "a gift is for a gift." If I want to sacrifice/dedicate something to a god(Des), I do it. I don't ask for anything in return, I just think, "Hey, I thought you might like this. Hope you do!" and move on.
As for giving a gift in exchange for a favor, I think the gift should somehow fit the god(Des). Plant something for the Vanir, burn something or bury meat/blowhole for the Aesir (though in retrospect I imagine any god(Des) would like food and booze dedicated/sacrificed to them...)
You got some good answers here. I agree more or less. An offering of ale or mead is good, if something really big was granted I sometimes give my own blood (a couple of drops poured over the altar stone. Don't exaggerate and endanger yourself.)
Apart from that, what we mostly give to the Gods is our belief and respect. Gods in turn give us the gift of interacting with our lives. Make the Gods a part of your everyday life and that's the biggest gift you can both give and receive.
Often times an offering of mead or a remembrance at summable is enough. Usually, however, Odin is one of those that, has a habit of deciding what he wants in return for any favor he has granted and avails himself of it at HIS convenience.
If you are asking about Othin, he is a fickle bastard, but also my all father. In the sayings of Har it is written, rightly ask but not over much for a gift begets a gift. The gift is not a physical one but acts of right and Reed. Othin asks of you what you ask of him, it can be a dangerous proposition, just ask a berserk. Yet, it can be related to what and how you ask. If you are being foolish he and the others will let your words punish as well as uplift. There is always an up and a down.
In Asatru, it's a gift for a gift. So if you ask a god like Odin for help and he does
