As bertoni says, I think the situation you're referring to is when an entire phrase modifies a noun, and the subject of that phrase is marked with の instead of が.

足が速い人
足の速い人

人がよく来る映画館
人のよく来る映画館

I think it's mostly a matter of preference and either is acceptable.

In classical Japanese, が and の both had pretty much the same function (が was also used for possessives) and that may have something to do with it. One other theory I've always held, though I don't know if there's any truth to it, is that the subordinate clauses could be broken down two ways and the の conveys that everything after the の is something that "belongs" to the thing before the の:

[足の速い]人
足の[速い人]

[人のよく来る]映画館
人の[よく来る映画館]

Yes, I'd say that's correct.

From DOBJG's explanation of relative clauses, p. 378:

3.  No can also be used to mark the subject in relative clauses, as in (6).

     (6) ジョン が／の 食べたステーキ
          (the steak that John ate)

    No cannot be used in place of ga, however, if the subject of the sentence is followed by a noun, as in (7a).  If no is used in this construction, the meaning of the sentence changes, as seen in (7b).

     (7)  a.  トム が フットボールの切符 を あげた女の子
                (the girl to whom Tom gave a football ticket)

           b.  トム の フットボールの切符 を あげた女の子
                (the girl to whom (someone) gave Tom's football ticket)