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I'm not sure if this is actual keigo, or just a polite form of adjectives.
Anyway, there are several that we're all familiar with that are still used
today.

>   * はやい → おはようございます
>   * ありがたい → ありがとうございます
>   * めでたい → おめでとうございます
>

There are a couple of others I've seen in textbooks at some point as well:

>   * たのしい → たのしゅうございます
>   * おいしい → おいしゅうございます.
>

I know that I've seen different rules for how to form these depending on the
ending. We can see that the first three end in **〜ai** that it becomes **〜ou
gozaimasu** , and the two **〜shii** ones become **〜shuu gozaimasu**.

However, I don't know the hard rules about how to form them based on their
endings. Furthermore, I'm even more confused about when to add the starting お
(はやい adds it, but ありがたい doesn't).

I know these forms aren't used much these days (except "set" phrases like the
first three), but it's a piece of knowledge that has eluded me over the years.
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conjugations keigo adjectives i-adjectives
～うございます - keigo い-adjectives
I will answer the two questions separately.

### How to make the form of i-adjectives before ございます

Grammatically はよう, ありがとう, めでとう, たのしゅう, おいしゅう in these examples are called ウ音便
(うおんびん) of はやく, ありがたく, めでたく, たのしく, おいしく, respectively. 音便 (おんびん) means the
form modified for easy pronunciation.

The actual form of ウ音便 of an i-adjective depends on the vowel before く in the
-く form (or equivalently, the vowel before い in the dictionary form).

  * If the vowel is -a, ウ音便 is -ou: はやく→はよう, ありがたく→ありがとう, めでたく→めでとう. This -ou is pronounced as a long vowel /oː/.
  * If the vowel is -i, ウ音便 is -yuu: たのしく→たのしゅう, おいしく→おいしゅう.
  * If the vowel is -u, ウ音便 is -uu: さむく→さむう.
  * (If the vowel is -e, ウ音便 should be -you, pronounced as a long vowel /joː/, but see below.)
  * If the vowel is -o, ウ音便 is -ou: おそく→おそう. This -ou is also pronounced as a long vowel /oː/.

(I could not think of any i-adjective whose dictionary form ends with -ei, and
the only adjective of this form in
[EDICT](http://www.csse.monash.edu.au/~jwb/edict.html) is [執念い
(しゅうねい)](http://dic.yahoo.co.jp/dsearch?enc=UTF-8&p=%E5%9F%B7%E5%BF%B5%E3%81%84&dtype=0&dname=0ss&stype=0&pagenum=1&index=109091300000),
which I had never heard of. The pattern above is estimated from the discussion
below.)

These forms of ウ音便 are the result of two sound changes.

  * In the first step, く was replaced by う: はやく→はやう, たのしく→たのしう, さむく→さむう, おそく→おそう.
  * In the second step, the -au became -ou and the -iu became -yuu: はやう→はよう, たのしう→たのしゅう. (This sound change is not limited to ウ音便 of i-adjectives, and in other words, -eu became -you with pronunciation /joː/. This is why I wrote the ウ音便 of i-adjectives ending with -ei should be -you.)

### Why sometimes with the prefix お and sometimes without お?

I can explain this only partly.

The prefix お shows respect. おはようございます literally means “You are early,” and
おめでとうございます literally means “(what happened to you) calls for celebration.” In
these cases, the adjectives describe something about “you,” hence the prefix
お. On the other hand, in the sentence たのしゅうございます “I have fun,” the adjective
たのしい describes the state of “me,” hence no prefix お.

However, this argument suggests ありがとうございます should have the prefix お because it
literally means “What you did is hard to exist,” and the subject of ありがたい is
the action of “you.” I do not know any explanation why ありがとうございます is not
おありがとうございます.
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answer