As a title, I have found to meanings for 自个: "since a" or "oneself", what is correct and how to translate it?
Zhongwen.com illustrates the different phrases one can put together with the character 自 (zì). Note that "oneself" and "since then" are valid possible meanings.
Jukuu.com provides many sample sentences for both of these phrases, here are two:
She was humming a tune to herself.
(note that 從 cóng is simplified here as 从)
He hasn't been home since last year.
oneself, by oneself
不要只顾自个儿。Don’t just think about yourself.
自个儿 is the a colloquial way to say 自己. On top of my head, I couldn't see any example where they are not interchangeable. The only difference between them perhaps is just that 自己 can be used in formal context while 自个儿 is rather colloquial.
Literally the word is composed with "self" "one" (儿 is just there to mimic the colloquial adjustment of pronunciation). So it's simply means self and one which could be understood as oneself, by oneself, oneself's etc. depending on the context.
It's can be used as a pronoun alone like the example given above.
More than often it would follow another noun part, subject or object and etc, to stress that it's object carry out the action alone. This can also make the reference more clear since 自个儿 itself only means oneself buy doesn't specify who the one is.
For example: 你自个儿 means "you and oneself", which in turn means "yourself".
So when you encountered 自个儿, just keep in mind it means that the object it refers to is finishing the sentence by itself and add appropriate adverb (by adding alone, by oneself) or change the pronoun (you->yourself) that fits other parts in your translation.
ABC dictionary defines it as:
oneself; by oneself
Basically it's the same as 自己 - it's just a regional thing.
Here's what Baidu says, with some example sentences.