> Usually words that end in 는 are "subjects" or "adjectives".

It's probably worth distinguishing between two completely different '는' you will see (i.e. they are homonyms - they look the same, but are different constructions with different meanings).

 1. The first one is one form of the [은/**는** 'Topic marker'][1]. This marks the topic, or the thing that you want to draw attention to in the sentence. (Note that while this is often used on subjects, it's not the 'subject marker' - that's 이/기.

 2. The second one is the present tense version of the [-ㄴ/은, **-는**, -ㄹ/을 family of adnominal endings][2]. 'Adnominal' means that they create a form of the verb that modifies the following noun, as explained further in [gaeguri's answer to your related question][3].

In the first meaning, you will see 는 attached to nouns. In the second meaning, you will see it attached to verbs. As 되다 is a verb, we know that this is an 'adnominal' 는 - it doesn't mark the topic, rather it creates a form of the word 되다 that can modify a following noun - in this case, '사람'.

So we could break the sentence down like this:

> 친구가 되다

 *to become a friend*

> 친구가 되는 사람

 *a person who becomes a friend*

(see how the 는 here creates a participle that modifies the following noun).

> 친구가 되는 사람이고 싶다

  *I want to be a person who becomes a friend*

 (or as you say, " want to be a person that can become a friend.").

  [1]: https://en.wikipedia.org/wiki/Topic_marker#Korean:_%EB%8A%94/%EC%9D%80
  [2]: https://123learnkorean.wordpress.com/2009/07/04/%E3%84%B4%EC%9D%80-%EB%8A%94-%E3%84%B9%EC%9D%84-adnominal-suffixes-korean-grammar/
  [3]: https://korean.stackexchange.com/a/3294/64