Not one, not two, but three "honorific" characters at the beginning of this word. It's pronounced omiotsuke and it's a polite way of referring to miso soup (served as part of a meal).

Start from the last two characters, 御付, (otsuke). This is an example of 女房詞 (nyouboukotoba), literally "wife words" -- euphemistic or polite-ified words for (mostly) food, cooking and related matters, invented by the women of the imperial court about half a millennium ago. They quickly spread to the general population, as "noble" language tends to, and some of them remain to this day: for example, ohiya for a glass of water (from the hi(y)- stem, meaning "cold") and onara for a fart (from narasu, meaning "make a noise").

There were also a lot of nyouboukotoba that ended in moji, "characters". This was more or less equivalent to the English "the (X)-word". For example, for sushi you could say sumoji -- "the su-word". These moji words could even take grammar-bearing suffixes: for "embarrassed" (hazukashii) you could say hamojii -- "the ha-word-y". Sadly, the moji words have all but vanished today.

So, otsuke is a euphemism for "soup that comes with a meal", by way of honorific prefix o- and the verb tsukeru -- "attach", "include with", etc.

The first half of the word, omi (御御), is -- if the same as the omi that is usually written with those characters -- a not-uncommon chunk which comes from ohomi (大御): "great" + mi, another honorific prefix. If this etymology is correct, then although 御御御付 is written "honorable honorable honorable attachment", it might come from a phrase that only means "greatly honorable honorable attachment".

On the other hand, though, some etymologists claim that 御御 are just ateji for this particular omi. Its real roots, they say, lie in the much more obvious omiso, i.e. honorific prefix + miso (soup).

I find this explanation more convincing, but at the same time I understand the impulse that led the first person to write the word 御御御付 instead.

Final notes:

1. The o prefix itself is said to derive from ohomi or ohomu as well, which as far as I can tell makes m- the original Japanese "honorific sound". Although, to be perfectly honest with you, the etymology of these honorific prefixes is a fever swamp, and it isn't rendered any easier by the fact that the character 御 can represent virtually any of them, including go-/gyo-, which came from Chinese.
3. The usual caveats about folks not consciously thinking all this "honorable" stuff, of course, apply.