First, I think a little more context may be needed to appropriately answer your question since it seems whoever you've been talking to has what I would consider to be a non-standard usage of it.
That being said, here's my best cut.
In standard usage, the phrase "not uncommon" typically means something to the effect of "more frequent than uncommon".
That is, it occurs too frequently to be considered "uncommon", but not necessarily so frequently as to be labeled "common".
However, in certain contexts where things are partitioned using discrete, well defined terms "uncommon", "rare", etc. it would be conceivable that they would mean "anything other than uncommon".
The one example that jumps out at me is RPG games like World of Warcraft where item quality is labeled using terms like "rare", "uncommon", "common", etc.
In a case like this, saying that something is "not uncommon" would potentially allow for something labeled as "rare" as well as something labeled as "common"
english.stackexchange.com/users/2007/Dusty
"not uncommon" is not necessarily the same as common.
Compare the following examples: 1-0 is a common final score in soccer (it happens all the time) 8-7 is an uncommon final score in soccer (it rarely happens) 5-0 is not an uncommon final score in soccer (it doesn't happen all the time, but it's not rare either).
english.stackexchange.com/users/212/b.roth
To my mind, it is a shaded meaning, not quite common, but not rare. "
Not unkind" I would see as much colder than kind, but not actively unkind.
english.stackexchange.com/users/2509/Remou
Not really.
It means just what it says it is.
It's an attempt at understatement.
To baldly say something is common seems to some people too forthright.
And, to be fair, there may be a wide gap between what is uncommon and what is common.
If the subject is sunny days in Seattle, you may say they are uncommon.
But Seattle averages about one day in six with sunshine.
So you could say, with some legitimacy, that they are not uncommon since they are experienced with some regularity.
Still, you wouldn't really say they were common.
Nevertheless, such constructions are anathema to some champions of clear writing.
I remember Eric Blair (George Orwell) mocking them in one of his essays on the English language, positing a sentence like (I'm paraphrasing here) "The not unbrown dog trotted not unslowly along the not unrough road, finally making a not unsmall leap over the not unhigh fence."
Or something like that.
I was an undergrad at the time, so memory dims.
I wish I could remember the name of the essay.
english.stackexchange.com/users/2303/Robusto
Describing something by negating its opposite is a way of countering expectations.
As in: "You might expect this never happens, but it is not uncommon."
Or: " Not bad... for a girl. "
english.stackexchange.com/users/990/Doug
This is an example of a common (and ancient) figure of speech called litotes .
Sometimes it conveys no extra meaning (and gets ridiculed as an affectation) but in many cases it does have a distinct meaning.
english.stackexchange.com/users/547/Colin Fine
When somebody uses the phrase "not uncommon", do they mean "common" because of the double negative?  I was recently informed that "not uncommon" could imply that something was anything but uncommon (e.g. rare), whereas I thought that  rare  would be a subset of  uncommon  and that saying it was not uncommon must mean it is common.
Does "not uncommon" mean "common"?
english.stackexchange.com
<meaning>
english.stackexchange.com/questions/6124
english.stackexchange.com/users/2570/staticbeast
The answer to this question will require a little more information about the context. However, this construction is known as a litote and is a form of understatement. "Not uncommon" is not necessarily the same as "common" and in fact means that something is not common but not rare either. There is insufficient context to answer the question. However, "not uncommon" can mean "more frequent than uncommon". Others feeling the meaning is clear that something is not a common occurrence. Some think that it can be ambiguous, others feel that using the word can be too "forthright".
The answer to this question will require a little more information about the context. However, this construction is known as a litote and is a form of understatement. "Not uncommon" is not necessarily the same as "common" and in fact means that something is not common but not rare either. This is an example of a common (and ancient) figure of speech called  litotes . "Not uncommon" is not necessarily the same as common. It is a shaded meaning, not quite common, but not rare. Describing something by negating its opposite is a way of countering expectations.
"Not uncommon" is not necessarily the same as common, and it typically means "more frequent than uncommon", often used as an attempt at understatement. The answer to this question will require a little more information about the context. However, this construction is known as a litote and is a form of understatement. "Not uncommon" does not mean the same as "common" and in fact means that something is not common but not rare either.
It's an attempt at understatement. This is an example of a common (and ancient) figure of speech called litotes .
"Not uncommon" usually means "more frequent than uncommon" It seems ambiguous, meaning not very common, but not rare/ The meaning is clear. There is also the meaning of "common" which can appear to forthright. It sometimes delivers no extra meaning.
"Not uncommon" is not necessarily the same as common. It is a shaded meaning, not quite common, but not rare. Describing something by negating its opposite is a way of countering expectations. This is an example of a common (and ancient) figure of speech called  litotes .
The answer to this question will require a little more informaton about the context. "Not uncommon" is a way of saying that something is not rare but not common. "Not uncommon" does not mean the same as "common". This is an attempt at understatement. This is an example of a litote.