Hebrews 5:8-9
8 Though he were a Son, yet learned he obedience by the things which he suffered;
9 And being made perfect, he became the author of eternal salvation unto all them that obey him;
Verse 9 above is used to teach that Jesus will only give Salvation to those who obey Him for their entire lives, to an unspecified degree.
Is this a valid interpretation? Let's look at two views on this verse from the Free Grace perspective.
The first view, and the view that I currently favor, is that this is an instance of "obedience" being used to refer to the "obedience of faith", also called "obeying the Gospel", as in the following passages:
Acts 6:7
7 And the word of God increased; and the number of the disciples multiplied in Jerusalem greatly; and a great company of the priests were obedient to the faith.
Romans 1:5
5 By whom we have received grace and apostleship, for obedience to the faith among all nations, for his name:
Romans 16:26
26 But now is made manifest, and by the scriptures of the prophets, according to the commandment of the everlasting God, made known to all nations for the obedience of faith:
2 Corinthians 9:13
13 Whiles by the experiment of this ministration they glorify God for your professed subjection unto the gospel of Christ, and for your liberal distribution unto them, and unto all men;
2 Thessalonians 1:8
8 In flaming fire taking vengeance on them that know not God, and that obey not the gospel of our Lord Jesus Christ:
1 Peter 4:17
17 For the time is come that judgment must begin at the house of God: and if it first begin at us, what shall the end be of them that obey not the gospel of God?
Romans 10:16-17
16 But they have not all obeyed the gospel. For Esaias saith, Lord, who hath believed our report?
17 So then faith cometh by hearing, and hearing by the word of God.
All of the above are references to the imperative given to believe (John 8:24, Acts 3:19, Mark 16:15-16, etc.). If one believes, they have obeyed that imperative, and so faith is in that sense "obedience", and it can therefore be said that someone "obeyed" the Gospel.
Therefore, in this view, the verse in question from Hebrews is a reference to faith, and "eternal salvation" refers to the Eternal Life that one receives by faith (John 3:36, 6:47, etc.).
The second view of this verse from the Free Grace perspective highlights that all of the uses of the word "salvation" in Hebrews (Hebrews 1:14, 2:3, 2:10, 6:9, 9:28, 11:7) can be understood as referencing becoming Christ's partners in rulership in the eternal Kingdom to come. The word "salvation", then, is being used to refer to being delivered to a higher experience in that Kingdom.
This is a very compelling understanding of the verse as well, because of the epistle's constant emphasis on the future reigning of the believer with Christ (Hebrews 1:14, 2:1-8, 3:1-4:11, 6:11-12, 9:14-15, 10:34-36, 12:1-10, 12:28-29, etc.), and the almost universal twisting of all of such passages to refer to initial regeneration is why so many struggle with Hebrews. When we begin to see all these verses as originally intended - as motivational in intent (Hebrews 13:22), rather than threatening - the book of Hebrews falls into place.
In either case, if the passage in question were teaching that in order to be saved from Hell, one had to live in obedience to Jesus to some unspecified degree for their entire lives, we would need to abandon the idea that Salvation is not of works (Ephesians 2:8-9, Titus 3:5), or adopt the impossible position that obedience doesn't take work, in which case, obviously, none of the warnings in Hebrews would need to be given in the first place.
We would also need to reject that the New Birth takes place at the moment of faith, or really, at any point before death, for the myriad of reasons discussed in this article. However, the Bible is not only unequivocal in stating that the New Birth happens at the moment of faith (John 1:12, Galatians 3:26: 1 John 5:1), but the statements made about the New Birth, as discussed in the aforementioned article, make it clear that anyone who has the New Birth is going to live forever with the Lord.
This passage in question, like the rest of the book of Hebrews, is nothing to fear. Whichever position one adopts fits well into the Free Grace position - either as an affirmation of Eternal Life by faith in Christ, or a motivator for the reward of the inheritance (Colossians 3:24) that the New Testament focuses so heavily on.