"These things" is a reference to the letter that John is now drawing to a close; he is echoing the sentiments with which he began the letter:
2 (For the life was manifested, and we have seen it, and bear witness,
and shew unto you that eternal life, which was with the Father, and
was manifested unto us;)
3 That which we have seen and heard declare we unto you, that ye also
may have fellowship with us: and truly our fellowship is with the
Father, and with his Son Jesus Christ.
4 And these things write we unto you, that your joy may be full. (1 John 1:2-4)
John opens & closes his letter by explaining that he is writing to affirm the faith of his audience and to teach them of eternal life.
Does "these things" imply something additional to believing on the name of the Son of God?
"These things", both in 1:4 and in 5:13, is a reference to the entire letter; it is not a reference to a checklist.
John testifies of what he knows and wants his audience to know.
When we say I know am saved because I have believed, how do we know that we have truly believed?
Through the Holy Ghost.
Compare 1 John 2:27
But the anointing which ye have received of him abideth in you, and ye need not that any man teach you: but as the same anointing teacheth you of all things, and is truth, and is no lie, and even as it hath taught you, ye shall abide in him.
with John 14:26
But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you. (John 14:26)
Both speak of the role of the Holy Ghost to teach us and make truth known; see also John 16:13
Howbeit when he, the Spirit of truth, is come, he will guide you into all truth
John further establishes that our knowledge of truth comes through the Holy Ghost in 1 John 3:24
And he that keepeth his commandments dwelleth in him, and he in him. And hereby we know that he abideth in us, by the Spirit which he hath given us.
In the passage above John also speaks of the importance of living the commandments in order to come to know of their surety, in keeping with the teaching in his Gospel:
If any man will do his will, he shall know of the doctrine, whether it be of God, or whether I speak of myself. (John 7:17)
"These things" refers to the letter; John explains at the beginning and end of the letter why he is writing.
John teaches that we come to know divine truth through the Holy Ghost, and that living Christ's teachings in essential to coming to know of their truth.