Friends

"I'd like to be your friend," is OK.
"I'd like to be a friend with you," does not sound like a native speaker said it.
"I'd like to be a friend of yours," sounds OK. You'd like to join those who are already my friends.
"I'd like us to be friends," sounds like maybe the person refered to by "you" might not want to be friends and you're trying to convince her/him.

I agree.
I saw your position on "I could care less", and I must say THANK YOU. I have been trying to convince my friends of this and no one listens!. When I hear it, I want to hurt myself. "I couldn't care less" It's not very difficult to add an n't. People.

Friends is plural in this sentence because it describes a relationship between two people.

Everyone always tells about their friends, they'll be 'Friends Forever,'but how often does that last? You might be best friends one year,pretty good friends the
next year, don't talk that often the next year,
and don't want to talk at all the year after that.

Remember, everyone needs a friend. Someday you might feel like you have no friends at all, just remember this e-mail and take comfort in knowing somebody out there who cares about you and always will.

Remember, to never lose touch with those friends you've made here, because you have all changed and grown enormously together, and that is something very sacred to be shared.

Remember, you are here for a short while, time flies before you realize it, so make it last, make it memorable, make it the best time of your life, the best memories that you can carry with you for the rest of you days, Remember, to love the ones you love, life isn't forever.

Remember, to love your friends, whether they come, go, love you or hurt you.

Stephanie missed the point.

There is a common occurance in any language where two phrases or cliched ideas become one because they are so closely related.
In this case, "friends" has unwittingly and inadvertantly become an adjective. When a culture becomes used to one version of a word, it tends to 'bleed' into other uses. At one point the phrase was probably "Are we friends? Yes, we're friends." People walked around town saying "Hey everyone, we are friends!" That subtle conditioning of using the plural noun form of the word all the time made it a "brand name" rather than a noun and thus the functionality of the word is ignored. Example: Do you have a Kleenex? Yes, I have three Kleenex, but I don't have five Kleenexes. See what I mean? The brand-name phenomenon trumps the grammar.

I would say it's the plural thing
If you are friends it's because you are THEIR friend (1) and they are YOUR friend (2)

if you are their FRIEND (1) , you are singular. Although if someone is your friend, it goes without saying that you are their friend also. So it's always going to be a plural

I would have to disagree with elmtailye. The verb "be" is a linking verb that identifies the subject as equal to the verb. The singular "I" cannot be equated with the plural "friends." It would be proper to say "Let's be friends," as then it is a plural "us" with "friends."

I think it's pretty simple.

To stem from Jim's response, the suggestion of "I'd like to be a friend with you," in fact, sounds incorrect to a native speaker.

It is because you are stating that the two of you will be friends. Not a friend. Friends to one another, rather. You will be friends. That is what you want. To be friends. Not to be a friend.

To be a friend is one-sided. Like, go and be a friend to the ostracized kid. It implies a one-sided effort.

So most people proffer "to be friends", as in a mutual agreement to a relationship.

I think the "I like to be friends with you" is a corruption of the more classical (for lack of a better word) "I would like to befriend you" with the prefix be- denoting the action.

"Let's be friends" is a quite proper shortening of "Let's you and I be friends."

Actually, let's is a conjunction of let and us so "Let's be friends" would not be "Let's you and I be friends" because that is nonsensical. "Let us be friends" sounds more appropriate, ne?

Yeah, you're right. Sounds better that way. "Let's you and I be friends" is a childhood memory.