A new Victorian-set British novel begins sentences containing h-dropping with lower-case e's, as in " 'e took my money, 'e did." This seems incorrect but I'm at a loss to find the rule.
I don't think anybody would ever write
'twas the night before Christmas,
with a small 't' (although sometimes the apostrophe is dropped here). I've checked some websites, and they all use a capital 'T'. And the title of John Ford's play is usually written
'Tis pity she's a whore,
with a capital 'T'. So I would say the rule is to capitalize the initial letter even if the sentence starts with an apostrophe.
And here is an example from the year 1897 which not only capitalizes the E in 'E, but also capitalizes the A in 'Allelujahs:
‘'E was a clever un, 'e was, if you like. Fancy workin' the Vanguard into a pome, like they work 'Allelulah's into 'ymns.’
The punctuation at the beginning looks a little awkward, with a left quote followed by an apostrophe, but that's the way they did it.
I would follow the style where capitalisation is done as if the apostrophe is absent and the word begins with the first of the remaining letters.
'Fraid so. 'Nother drink?
I s'pose so. 'S not funny.
(examples taken from Larry Trask's Guide to Punctuation)
When it appears in the speech of a character, it just tries to show the accent of the particular character which deviates from the standard British one. There is no rule involved in the language spoken at that time. As to whether this is correct in terms of printing, I think there's nothing wrong with the lower case of the "e" here once the initial letter is dropped; the apostrophe marks the absence of "h".
Although commonly misstated, that is, stated in a simplified form such as "capitalize the first word of a sentence", the "rule" that applies in the circumstance you've described was originally a printing convention termed 'sentence case'. That convention has in modern times transferred, for the most part silently, to typographical media other than printing, for example, typographical web content presented via a browser rather than via the printed page.
'Sentence case', given full expression, is the convention that the first letter of the first word of a sentence is capitalized, while any remaining letters in the sentence are capitalized according to more specific rules pertaining to proper nouns, acronyms, etc. An apostrophe is not a letter, so a sentence beginning with an apostrophe has the first letter following the initial apostrophe capitalized, by convention. As explained in The London School of Economics and Politics Web Editors' Handbook,
Sentence case is a standard approach to the use of upper and lower case letters, mainly in titles and headings. It is used in publishing regardless of medium but commonly in newspapers and other print media. Adopting sentence case ensures consistency throughout a site. ....
The use of case (ie upper or lower) follows the normal rules of a sentence in the English language. Specifically, capital letters are used for the first letter of the first word; proper nouns; and abbreviations/acronyms. All other words are lower case.
(From The London School of Economics and Politics Web Editors' Handbook, "Best practice guide". Emphasis mine.)
Various deviations from the 'sentence case' convention may be encountered in publications using in-house style rules. Typically, however, the deviation will be in whether and when 'sentence case' is used for headings or section titles. Deviation from the convention for the example you've given, where an omitted letter (the aitch) is replaced by an apostrophe at the beginning of a sentence, is rare, although it may be encountered. Thus, the usual but not invariable practice is to use 'sentence case' for the given circumstance:
'E took my money, 'e did.
Historically, deviations from the 'sentence case' convention may be more common in examples drawn from earlier periods during the development of the 'sentence case' printing convention.
Your difficulty discovering the "rule" governing capitalization of a sentence with a sentence-initial apostrophe might have been the result of the "rule" being a type of "Purloined Letter" mystery: the "rule", more properly described as a convention, is in plain sight, and thus hidden. It is a basic convention of English typographical material, and finds expression in most style guides. For example,
Capitalize the first letter of a word that begins a sentence ....
(From the OECD Style Guide, page 66.)
Similar capitalization 'rules' are given in most thorough style guides.
Additional difficulty in discovering the "rule" may have arisen from a failure to recognize that apostrophes used to indicate a letter has been omitted are not themselves letters in the English alphabet, but rather are considered signs or marks.
2. The sign (') used to indicate the omission of a letter or letters, ....
["apostrophe, n.2". OED Online. March 2016. Oxford University Press. http://www.oed.com/view/Entry/9449 (accessed April 23, 2016). Emphasis in definition mine.]
Thus, an 'apostrophe' is contrasted with a 'letter'. Apostrophes are not part of the 26-character English alphabet:
I. An alphabetic character, and related senses.
1. a. A character representing one or more of the elementary sounds used in speech and language; any of the symbols of an alphabet used in written language.
["letter, n.1". OED Online. March 2016. Oxford University Press. http://www.oed.com/view/Entry/107545?rskey=GPGcAh&result=1&isAdvanced=false (accessed April 23, 2016).]
The pertinent part of the denotation of 'alphabet':
... esp. (with the) the set of 26 letters from A to Z used to write words in English ....
["alphabet, n.". OED Online. March 2016. Oxford University Press. http://www.oed.com/view/Entry/5693?rskey=TS1aBu&result=1&isAdvanced=false (accessed April 23, 2016).
In this case, you're dropping the H from a pronoun. I think the answer to this question lies in the way we treat proper nouns. When H-dropping a proper noun we never capitalize the letter following the H:
In My Fair Lady, Eliza Doolittle described the weather in three English counties: 'in 'artford, 'ereford and 'ampshire, 'urricanes 'ardly ever 'appen' ('artford = Hertford, generally pronounced as 'Hartford').
Additionally her pronunciation of Professor Higgins’ name comes out as ‘enry ‘iggins.
From the above, I would suggest that the correct way to H-drop the sentence "Henry stole my apple." would be as follows:
'enry stole my apple.
Which leads me to conclude that replacing the noun with a pronoun, from a grammatical perspective, follows suit:
'e stole my apple.
Welcome to EL&U. Since the sentence wouldn't begin with a small 'e' in normal prose, there's no reason why it should do so when it replicates speech. The ' stands in place of the missing 'H' at the beginning of the sentence and in place of 'h' before did. Some publishers may choose to print it as 'E took my money, 'e did, but I think there is a stronger argument for the version which you report.