In classic Greek, the word has an astounding variety of meaning:
I. in a form of sun-dial, the shadow of the gnomon, the length of which in feet indicated the time of day...
II. element
1. a simple sound of speech, as the first component of the syllable...
2. in Physics, στοιχεῖα were the components into which matter is ultimately divisible, elements, reduced to four by Empedocles, who called them ῥιζὤματα...
3. the elements of proof...
4. generally, elementary or fundamental principle...
5. ἄστρων στοιχεῖα the stars...
6. ς. = ἀριθμός, as etym. of Στοιχαδεύς, Sch.D.T.p.192 H.
Of these, I think we can eliminate I., which relates to the sun-dial. But we must also remember that the word would have continued to carry the connotation of the passage of time. Therefore the link between στοιχείων τοῦ κόσμου and special days in Galatians and Colossians less of a coincidence.
The author of 1st Peter seems to have had II. 5. (stars) or possibly II. 2. (the physics term) in mind:
But the day of the Lord will come like a thief, in which the heavens will pass away with a roar and the elements will be destroyed with intense heat, and the earth and its works will be burned up.
Since all these things are to be destroyed in this way, what sort of people ought you to be in holy conduct and godliness, looking for and hastening the coming of the day of God, because of which the heavens will be destroyed by burning, and the elements will melt with intense heat!—1st Peter 3:10-12 (NASB)
This is apocalyptic language that indicates the total destruction of the current state of affairs to make way for a new heavens, a new earth, and a new relationship between God and people. Destruction of the stars would fit naturally in the usual imagery of "the day of the Lord". I lean against the technical physics term as at the time fire was considered an element. It's also not clear if the audience would have conceived of water and air being "melt[ed] with intense heat". That leaves just one element under the classical system.
Meanwhile the author of Hebrews uses strictly the II. 4. sense:
For though by this time you ought to be teachers, you have need again for someone to teach you the elementary principles of the oracles of God, and you have come to need milk and not solid food.—Hebrews 5:12 (NASB)
My loose paraphrase would be:
You ought to be teachers, but you still don't know the first thing about the Bible...
That leaves Paul's two letters. We have two good interpretations of those passages. (One of them is mine and the other is in answer to my question, so I have possible bias on this point.) To me, Paul is following closer to the meaning from Hebrews than from 1st Peter, but I don't think he minds the connection to astrology and the passage of time. Thayer's Greek Lexicon makes the case against heavenly bodies:
Hence some interpreters infelicitously understand Paul's phrase τά στοιχεῖα τοῦ κόσμου, Galatians 4:3, 9; Colossians 2:8, 20, of the heavenly bodies, because times and seasons, and so sacred seasons, were regulated by the course of the sun and moon; yet in unfolding the meaning of the passage on the basis of this sense they differ widely.
And makes the case for a more philosophical readings of Liddell and Scott II. 3. and 4.:
The elements, rudiments, primary and fundamental principles (cf. our 'alphabet' or 'a b c') of any art, science, or discipline; e. g. of mathematics, as in the title of Euclid's well-known work...
The elements of religions training, or the ceremonial precepts common alike to the worship of Jews and of Gentiles, Galatians 4:3, 9... specifically, the ceremonial requirements especially of Jewish tradition, minutely set forth by theosophists and false teachers, and fortified by specious argument, Colossians 2:8, 20.
Summary
Paul uses phrase "the elements or principles of this world" in contrast to the principles of God's paradigm for the Church. Other connotations of the phrase are not unwelcome, but must be strictly secondary in order to follow his argument.
