What is does it mean to have an identity? This can be considered in different ways:
Personal characteristics such as age, date of birth and nationality
Personal interests, character and other internal aspects
What makes us the same person despite changes over time
How our consciousness gives us an identity and is essential to all conceptions of self-identity
In Locke's view self-identity is based on self-consciousness and memory. Self consciousness is what distinguishes us from animals who only have a sense of their body and not their self. Memory according to Locke is what allows us to have a sense of our self identity over time. Through our memories we connect the pervious self with the current self and allow our self-consciousness to confirm our self-identity.
A problem with Locke's view is that it does not easily accommodate forgetting memories over time and situations in which there is a false memory.
Hume's view is that we have fleeting memories and that these do not provide a reliable connection with our past self. Hume's view is that our thoughts, ideas and memories pass through our minds and do not have a long term existence and so do not prove our self-identity.
Hume does not see consciousness as helping confirm self-identity as he sees consciouness as being a series of thoughts, without a way to prove the connection between one thought and the next.
Hume considered that over time, objects undergo small changes and that they are no longer necessarily the same object.
For both Locke and Hume self-consciousness is important in considering self-identity. For Locke self-consciouness and memory provide confirmation of self-identity. For Hume consciouness does no truly exist, instead what we experience is no more than a series of thoughts passing through the mind.
The view of Locke provides a clear sense of self-identity and a clear link between our memories and changes over time. However, the ways in which memories can be forgotten and can be different from reality undermine the truth of this view.
The view of Hume appears more truthful as it accepts that memory cannot be relied on to define self-identity.