"This is Lord Henry, a good friend of mine"
What differences are there? It is just stylistic differences? Isn't? Can I freely use such construction instead of "my/our etc"? For example:
This is a house of ours .
I've met many friends of yours.
A teacher of mine said that...
And also such variant:
I've discovered much useful of yours.
In some situations, saying "my [x]" as opposed to "a [x] of mine" could imply that the subject of your sentence is one of a kind.
Example:
This is a house of mine.
Because you have used the indefinite article this implies that you have more than one house, as if you had said "this is one of my houses", which would be more explicit.
If you said instead:
This is my house.
Although this does not exclude the possibility that you own more than house, it does not imply it either, and I think there would be an assumption you only own one. If you said "this is my home" that assumption would be even greater, as people tend to refer to just one place as their home, even if they own multiple properties.
With your example that refers to "a friend", I would say that, unlike houses, there is more of an assumption that most people have more than one friend. Therefore I would say there is no discernable difference between the statements "this is my friend" and "this is a friend of mine".
In the case of teachers which you mention in another example I would say that if it was to be assumed you only had one teacher at any given time, saying "my teacher" would imply uniqueness and perhaps that you are referring to your current teacher, whereas "a teacher of mine" implies either that you currently have, or have had more than one.
As in a lot of situations, context is everything and you should consider what your choice of words says about your specific subject.
“My X” tends to imply that there is only one X. This can mean that you only have one X, or that it is clear from the context which X you are talking about. This is not absolute: depending on what X is and on the context, it may be strongly implied, or weakly implied, or even sometimes not implied at all.
For example, “this is my friend Lord Henry” sounds perfectly natural, because (presumably) you only have one friend who is called Lord Henry. On the other hand, “this is my old friend” sounds a little weird: it sounds like you have a single old friend (meaning, a single elderly friend). In this specific case, there's an additional issue with the adjective “old”. In “this is my old friend”, it could either mean “elderly friend” or “person who has been my friend for a long time”. On the other hand, “an old friend of mine” can only mean “a person who has been my friend for a long time”.
To my non-native (but fluent) ears, “this is X, a friend of mine” and “this is my friend X” and “this is a friend of mine” all sound natural. “This is my friend” only sounds natural if you've just been talking about that person, and it then means “this is the friend that we were talking about”.
Nonetheless, there are contexts where it's perfectly idiomatic to say “my friend” without implying that you have a single friend. For example, in the sentence “he's my friend, I won't let him down”, there is absolutely no implication that you have a single friend. I find this particular more idiomatic than “he's a friend of mine, I won't let him down”. I don't know why; maybe it's because “he's my friend” makes your connection closer and this is an important impression given the sentiment expressed by the sentence?
Similarly “a teacher of mine said …” is the natural way of saying that one of your teachers said something without specifying which teacher. “My teacher said …” either refers to the specific teacher that you have been discussing earlier (it's equivalent to “the teacher [of mine] we were discussing earlier said …”), or implies that you have a single teacher.
“I've met many friends of yours” could equally be “I've met many of your friends”. The first sentence takes “a friend of yours”, pluralizes it to “friends of yours”, and adds the determiner “many” to indicate that the number is more than a few. The second sentence takes “your friends”, and adds “many of” to indicate that the number is more than a few but less than all. The meaning of the result is the same.
“This is a house of ours” implies that you have several houses. “This is our house” implies that (like most people) you have a single house.
“I've discovered much useful of yours.” is not grammatical: it's missing a noun or pronoun at the core of the direct complement of “discovered”. I don't understand what you mean to say.
