It's a very common pronunciation of 'my' in the North of England (and some Australian accents).
According to Wikipedia:
Other areas of the North have regularised the pronouns in the opposite direction, with meself used instead of myself. This appears to be a trait inherited from Irish English, and like Irish speakers, many Northern speakers use reflexive pronouns in non-reflexive situations for emphasis.
Moreover, the vowel in 'me' (the one you're referring to) is shorter ([mi]) than that of the regular 'me' ([mi:]) and this pronuncation of 'my' is usually used where 'my' is unstressed.
The pronunciation of emphatic/stressed 'my' was [mi:] in Middle English (before the Great Vowel Shift) and its unstressed/unemphatic form was [mi].
The Great Vowel Shift was a vowel change (1400-1700) that shifted almost all the long vowels to diphthongs but it did not affect short vowels. So the pronunciation of emphatic 'my' shifted to [aɪ]* but the short vowel in 'my' [mi] in its unemphatic form remained unaffected in some accents (Northern British, Irish, some Australian accents etc).
*GVS shifted [iː] to [aɪ]. For example, the pronunciation of 'bite' was /bi:t/ before the GVS.
Edit: See JDeBP's comment:
It is important to stress that this is not a substitution of "me" for "my" as the question asserts. It is a pronunciation of "my", and can also be observed for many other words ending in "-y", including the word "many" in this very sentence.