It doesn't matter. In fact, I believe it's a question that only later Christians (anachronistically) bring to the text. The original audience would not have thought of being a believer (or not) in such a binary fashion. But I'll first indulge an argument for the rich man being a brother.
The rich brother
"The brother" ("ὁ ἀδελφὸς" / ho adelphos) is a nominative singular noun and "the rich [man]" ("ὁ πλούσιος"1 / ho plousios) is a nominative singular adjective, and both share the same verb, "let [him] boast" ("Καυχάσθω" / Kauchasthō). Thus "the rich" ("ὁ πλούσιος") appears to be a modifier of "the brother" ("ὁ ἀδελφὸς") which would be implied in v. 10. According to Davids, this has been the majority position of scholars:
Is ὁ πλούσιος a modifier of an understood ὁ ἀδελφός in parallel with v 9, or does one encounter a poor/humble brother and a rich non-Christian? Structurally the former alternative appears most likely, for the sentence demands that καυχάσθω be understood as the verb of v 10. In this case the wealthy Christian is instructed to take no pride in possessions or position, but rather to think on his self-abasement in identifying with Christ (i.e. repenting) and Christ’s poor people. This is how most scholars have interpreted the phrase (e.g. Adamson, Cantinat, Mayor, Mussner, Ropes).2
The rich man (regardless of whether or not he identifies as a brother)
Despite this being the majority position, other scholars note the frequent contrast of the poor remnant of Israel with the rich who oppress them in Jewish literature.3 I see such Jewish themes throughout James' letter. In fact, the only other use of "the rich [men]" ("οἱ πλούσιοι" / hoi plousioi) in James occurs in 2:6 where the rich are portrayed as oppressors of the poor man, and such polemics occur frequently. Perhaps more importantly, the Christian author of this text states that the rich man "will wither away" (μαρανθήσεται / maranthēsetai)—a dismal fate. Dibelius states:
Which interpretation does our saying demand? ... It seems to me that only the conclusion of the saying provides a decisive argument with regard to this question: the coming fate of the rich man is depicted with great elaboration, and not without satisfaction; and in fact it is only his downfall which is depicted! Nothing is mentioned which would compensate for this loss, no coming of the Kingdom is portrayed. Without any consolation for the rich man, the saying ends with “he will wither” (μαρανθήσεται). Downfall is the final prospect which Jas holds up to the rich man, and therefore he can only be speaking to him ironically when he refers to “boasting.” Hence, from v. 10 on the saying is intended ironically, and Bede is correct in his interpretation: “[The saying] is certainly spoken in derision, which in Greek is called ‘irony’ ”.... If, for the most part, commentators have been unwilling to have anything to do with this interpretation, this has usually been due to the fact that their interpretation has been based upon an understanding of “rich man” to mean a rich Christian. But the question “Christian or non-Christian?” is imposed upon this verse from the outside; Jas himself does not have it in mind. Consequently, this problem, which cannot be quickly resolved, can be postponed until the conclusion, and the meaning of the saying need not be made dependent upon its resolution. But furthermore, it has frequently been left out of account in the interpretation of this passage that the antithesis between poor and rich had a history already within Judaism, and that this history does not allow the interpretation of “humiliation” (ταπείνωσις) to mean some inner condition, such as the lowly status of being a Christian. Here the fate of the rich man at the great cosmic transformation is spoken of without any sentimentality, and it is precisely such harsh words about and against the rich to which we are accustomed in the “literature of the Poor” in Judaism....
[T]he much discussed question of whether the rich man is a Christian or a non-Christian ... can scarcely be solved by an exegetical route, for here one argument stands opposed to another: reading the passage without any preconceptions, one at first thinks that “brother” (ἀδελφός) must be supplied, but no trace of any allusion to a brotherly relationship can be found in the harsh words of vv. 10f. Therefore, it is by no means clear from the outset whether Jas has in mind a rich Christian or non-Christian. The history of the concepts poor and rich in Judaism indicates the reason for this lack of clarity: that portion of the people who called themselves “the Poor” viewed the powerful rich man as the lawless “transgressor.” It made no difference whether this transgressor belonged to the Jewish faith externally or not—in any case, he no longer belonged to it inwardly. Jas has taken over this model, and consequently he has said nothing about whether his “rich man” is to be sought within Christianity or not. Whom does he have in mind? In any event, it must be people who are spiritually alien to the pious community, for otherwise the conclusion of the saying could not read as it does. Therefore, he may have had in mind primarily non-Christians; but if he was thinking here of Christians as well, then these are people whom he considers no longer to be included in a proper sense within Christendom.4
Keeping "the main thing" the main thing
Davids wisely concludes his discussion of this issue in his commentary by stating,
Whether the rich person is a brother or not, the following proverbial statement [(i.e, v. 11)] invites us to consider the meaninglessness of wealth in the face of death.5
1 I've intentionally omitted the postpositive δὲ.
2 Peter H. Davids, The Epistle of James: A Commentary on the Greek Text, New International Greek Testament Commentary (Grand Rapids, MI: Eerdmans, 1982), 76–7.
3 For instance, cf. Sirach 10:21—11:1.
4 Martin Dibelius and Heinrich Greeven, James: A Commentary on the Epistle of James, Hermeneia—a Critical and Historical Commentary on the Bible (Philadelphia: Fortress Press, 1976), 85, 87-8.
5 Davids, 77.