Episode III reveals that Padme dies of a broken heart shortly after the birth of the twins. So how can her daughter, Leia, have memories of her more than 20 years later?

It's worth noting that the film doesn't get specific about the time frame between the birth and mother's death. It could be minutes, it could be hours. I think it's reasonable to say that the intent is less than 24 hours, since it is inter-cut with the re-construction of Anakin. By the time the Vader suit is on, Padme is dead and Anakin's medical needs seem pretty urgent. Suffice to say that regardless of a timeframe or minutes or hours, it seems fast for Leia to have made memories.

Yes, Leia's talking about Padme
I've seen it suggested that Leia is speaking of her adopted mother (the Queen of Alderaan, Bail's wife) rather than Padme. Let's look at the exchange...

LUKE: Leia... do you remember your mother? Your real mother?
LEIA: Just a little bit. She died when I was very young.

Asking someone if they remember their real mother is really gutsy, unless you've had previous conversations establishing that they are adopted. In that context, the "your real mother" clarification definitely means biological mother.

If they hadn't previously discussed it, Leia would have replied, "How did you know I was adopted?" (assuming Leia knows), or "What do you mean my real mother? I have only one mother."

Besides, if it's not Padme then this exchange has no dramatic meaning.

Images and Feelings
So, what does Leia claim she can remember?

LUKE: What do you remember?
LEIA: Just images, really. Feelings.
LUKE: Tell me.
LEIA: She was very beautiful. Kind, but sad.

It is certainly common for every day people to describe their earliest memories in terms of emotions and images. These impressions may or may not be accurate, but feel real. Typically specific lasting memories are not formed until a few years into life, but there are exceptions. This may or may not be true in the Star Wars universe.

While it is in no way critical to this discussion, nor can it be backed up by movie evidence, it is even possible that Bail at some point showed Leia a photograph of her biological mother. It is commonplace for the brain to mix memories of images with memories of the actual event.

Memories or Visions?
Yoda tells Luke, "Through the Force, things you will see. Other places. The future... the past. Old friends long gone."

The movies imply that Leia is Force sensitive. It is certainly possible that while growing up, Leia received visions of her mother through the Force. Visions of the past, precisely as Yoda describes.

Notice that Luke's Force vision of the future also came in images and feelings, "I saw a city in the clouds" and "They were in pain".

Pre-Birth Connection
It is also possible the Leia's impressions of her mother formed before her birth. What can the films tell us about this?

Yoda says of the Force, "Life creates it, makes it grow. Its energy surrounds us and binds us. Luminous beings are we, not this crude matter. You must feel the Force around you. Here, between you, me, the tree, the rock. Yes, even between the land and the ship."

Yoda is certainly not placing the burden of conscience or will, or even sentience, on the connections the Force will make. Life-to-life seems to be the only requirement. (Are rocks alive?) A mother-daughter binding in the Force existed in some form.

It is inconclusive as to whether the general connection in the Force of all things is enough for specific awareness or not. Certainly Padme is not aware of both twins. Then again, Padme is not portrayed as a Force user, so we can't necessarily expect her to feel these Force connections.

Qui-Gon explains to Anakin, "Midi-chlorians are a microscopic life form that resides within all living cells and communicates with the Force." Again, this is a clear description of the nature of the communication and connection in the Force to be an involuntary one, happening without need of will or consciousness.

It is safe to assume that as a feared offspring of Anakin, Leia has a midi-chlorian count that is at least above average if not exceptional. In the months Leia was in the womb, there would have been a lot of Force-based communication and connection with her mother, whether Leia was aware of it or not, leaving cellular, midi-chlorian, and even "real" memories of Padme.

Conclusions
Exceptionally early memory-making, later-in-life visions from the Force, or pre-consciousness connection through the Force are all viable reasons why Leia would have formed lasting memories of Padme in the minutes or hours they shared in a medical room together.

This isn't a plot hole.

Beyond What we "Know"... My Thoughts
My personal opinion cannot be backed up from the films, so I leave to you only as food for thought.

I believe that the highly Force sensitive twins were abnormally conscious and aware even in the womb, and that Leia's memories were more than passive, but were part of a real relationship formed before birth.

This line of thought is endorsed in the Star Wars expanded universe. In Dark Empire, Leia's unborn twins (Jacen and Jaina) lend their Force power to their mother, to help defeat Palpatine one last time.

Why not Luke?
I can hear many of you asking... OK, so why doesn't Luke remember? If you're the father of more than one child, you'll need little convincing that each sibling develops very differently... but perhaps I will tackle in more depth in the future.