This chapter is primarily answering questions asked of Paul by the church in Corinth.
Now concerning the matters about which you wrote:... 1 Cor 7:1
Some of those questions had to do with the state of marriage once the new birth enters in to the equation;
Verses 1-6 treat the matter of conjugal responsibilities within an existing marriage (permanent marital continence is strongly discouraged) .
verses 7-9 treat marrying as a protection against carnal desire for those eligible (the ability to remain celibate is from God, not an act of human will)
verses 10-11 prohibit divorce (and remarriage) and I think the following verses indicate that these two are regarding marriage between believing spouses.
verses 12-16 treat the matter of this new faith as a potential division within an existing marriage and the verse in question mentioning children is in the context of Paul responding to questions regarding the marriage dynamic where one spouse has come to faith and the other has not. It is not explicitly given that the state of children within this sort of a marriage was a part of the original questions, however it is reasonable to assume that it may be the case. It is also reasonable to read the particular phrase,
Otherwise your children would be unclean, but as it is, they are holy. (v. 14b)
in context, as a parenthetical explanation of the principle being addressed in the beginning of the same verse.
For the unbelieving husband is made holy because of his wife, and the unbelieving wife is made holy because of her husband. (v. 14a)
In other words, "If the principle of the unbelieving spouse being made holy by the believing spouse were not true then your children would not be holy but, as it is, they are."
This doesn't show us the difference that Paul intends to impart by using the terms 'unclean' and 'holy' but it does serve to illustrate that there is a participation in the 'holiness' (whatever that is) of the one by another.
But then in v. 15-16 it becomes clear that an unbelieving spouse being 'made holy' by the believing spouse is not the same as being 'saved':
But if the unbelieving partner separates, let it be so. In such cases the brother or sister is not enslaved. God has called you to peace. For how do you know, wife, whether you will save your husband? Or how do you know, husband, whether you will save your wife? (v. 15-16)
Here we have the believer/unbeliever couple and the believing spouse (who is currently making the unbelieving spouse holy in the marriage) may let the unbelieving spouse go if they wish to leave BECAUSE the unbeliever's final state as regards salvation and the believer's role in that salvation is unknown.
If the holiness imparted to the unbelieving spouse is not equated with salvation then the holiness imparted to the children according to the same principle should not be either. Therefore many of the attempts to explain what Paul means here are erroneous because, in their reasoning, holiness is confused with salvation.
So then, what meaning do these words, 'unclean' and 'holy', try to impart in this context if not salvation? The Hebrew word ko-desh (Holy) carries the meaning of being separated or set apart. We read into it a hinting of moral uprightness (which uprightness certainly should be a part of being set apart) but this is not intrinsically necessary in the definition; a plate, an animal, or an article of clothing can be Holy unto the Lord, that is to say dedicated for His use.
In reference to God it means that there is absolutely no other like Him; He is creator of all and utterly unlike His creation. In reference to believers it carries the meaning of being called out from one thing unto another thing and very often the transition is from uselessness in the Kingdom to usefulness. This all hearkens back to the system of the temple and the priests where certain items (and individuals) were acceptable for use in service to God. Some of those things were only ever to be used for that one purpose (they were always clean for use). Others could be used if certain cleansing rituals were undergone (they were available and could be cleansed). Still others were not allowed (they could not be cleansed).
Since the verse in question (1 Cor. 7:14) does not equate holiness with salvation and the terms holy and unclean can be shown to indicate availability for divine usage, Paul's point regarding these children might be that as long as there is at least one believer in the household the others in the household are thereby made available for cleansing. They are made holy in the sense that they are made available (to be cleansed and used) by their proximity in relationship to the believer (who is cleansed and useful) as opposed to being unclean (and not cleanable).
The final verse (16) makes sense in this light because the believing spouse is encouraged neither to expel nor retain the unbelieving spouse but rather to allow staying or going as they desire BECAUSE the believer can not know whether or not their relationship will result in salvation.
To further the idea that "holy" can apply to children prior to their coming to personal belief we have:
And when the time came for their purification according to the Law of Moses, they brought him up to Jerusalem to present him to the Lord (as it is written in the Law of the Lord, “Every male who first opens the womb shall be called holy to the Lord”) - Luke 2:22-23
Holy here is the same as holy in 1 Corinthians 7 and carries the meaning of separated entirely for the use of the Lord. Therefore, children in a marriage where only one spouse is a believer are thus holy (separated) unto the Lord. They will still need to be born again by grace through faith.