Brooks' and Winbery's Syntax of NT Greek (Lanham, MD: Univesity Press of America, 1979) points out that ὁ Λόγος in Jn 1:1c is the subject nominative within the construction since that noun phrase has the article while the preverbal anarthrous PN does not (page 78).
Of course there are exceptions to the aforesaid general rule, as Richard A. Young shows in his Intermediate NT Greek: A Linguistic and Exegetical Approach (Nashville, TN: Broadman and Holman, 1994) on pages 64-65. Nevertheless, I think it's safe to conclude that ὁ Λόγος is the subject nominative in 1:1c.
There is a question about whether one could rightly conclude that the articular occurrence of Λόγος in 1:1c necessarily signifies a person over against a non-personal entity (especially in view of the fact that the article can be and is used in the NT to describe impersonal objects). But there are possibly other indicators in the context that suggest ὁ Λόγος is a person.
Jn 1:1b declares that ὁ Λόγος was πρὸς τὸν θεόν. A number of grammarians and commentators believe this part of the verse describes the intimate relationship between ὁ Λόγος and τὸν θεόν.
1:9-14 also indicates that John delineated the features of a person. As he writes in 1:14:
"So the Word became flesh and resided among us . . ."
Additionally from Young (page 101): "John writes that the Word was with God (John 1:1 acc.). Harris (1978:1205) suggests that the πρὸς in John 1:1 refers to active communication rather than passive association."
Lastly, ὁ Λόγος is masculine: so is αὐτοῦ and μονογενοῦς, which doesn't necessarily mean the Word is a person, but the gender of nouns and pronouns along with the literary context definitely affects how we translate Bible verses.