According to Protestants, what is the biblical basis of Jesus being "one substance with the father"? The only place in scripture where the relationship of the substance of Jesus with the substance of God that I can find in the scriptures is To the Hebrews:1:3 which actually says that they are of similar substance:


  Heb 1:3  He is the radiance of the glory of God and the exact imprint
  of his nature, and he upholds the universe by the word of his power.
  After making purification for sins, he sat down at the right hand of
  the Majesty on high,


So where did they come up with the notion, according to "sola scriptura" Protestants?
Introduction

In language, words convey a meaning or idea. Much like the word Trinity conveys a theological concept that most protestants believe is found in scripture, the word "Trinity" itself is never found. This word was coined to describe an idea which previously had no accurate descriptor. 

Similarly, while the word ὁμοούσιος (homooúsios) which means in greek "Of similar substance" may not appear in scripture, the idea does. And this is not an altogether uncommon occurrence with words like transubstantiation, rapture, consubstantial, predestination and many other theological terms not actually being found in scripture that instead may (or may not) be found as concepts only.

Background: The Arian Controversy

So where does this word come from?  As it turns out, while the idea may come from scripture, the term that was selected to encompass this idea in the early church was a response to a specific teaching being most famously made by a priest named Arius. Arius taught that Jesus was a creation, stating


  If the Father begat the Son, he that was begotten had a beginning of existence: and from this it is evident, that there was a time when the Son was not. It therefore necessarily follows, that he had his substance from nothing.


In short, Arius was teaching that Jesus was ὁμοιούσιος (Homoioúsios) or "of a similar substance" to God. These teachings began to drive a rift in the Church, so emperor Constantine, who had legalized Christianity in the Edict of Milan convened a council to settle the matter.

Concerns of the Council

The use of the root -ὅμοιος (oúsios) clearly had philsophical/theological significance within Roman culture, which was undoubtedly influenced by platonic discussions of metaphysics and creatio ex nihilo, creatio ex materia, and creatio ex deo. In other words, this council was discussing how to formulate their theology in the context of the conversations and epiphanies of Roman philosophers. But, to extend the allegory of Plato's Cave, the terminology used - ὅμοιος (oúsios) is merely a shadow cast by ideas that are in scripture - the jargon are not the source of the idea itself.

For modern readers the question then is: Did Jesus claim to be


The same as God (The same substance)
Something like God (Similar in substance)
Nothing like a god or God at all (Of an altogether different substance)


And accordingly, within the council there were three major viewpoints:


ὁμοούσιος (Homooúsios), the position of Athanasius of Alexandria, writer of the Athenasian Creed which was similar to the Nicene Creed. He and his followers adhered to the prevailing view of the council which was reflected in their final statement, the Nicene Creed.
ὁμοιούσιος (Homoioúsios, note the one letter difference) (True Arianism) and ὅμοιος (hómoios) (the Acacians sect of Arianism). †
ἑτεροούσιος (Heterooúsios), the position of The Anomoean sect of Arianism


As you can see, the three viewpoints correspond to the three possible conclusions on the deity (or lack thereof as it may be) of Christ. Essentially, this just comes back to the very basic question of the Trinity and the question "Who was Jesus and who did he claim to be". As such, I will not re-hash here the discussion of whether or not the Son is Triune and if there is a Biblical basis for this as this topic has been well covered here on Christianity.SE.

The aftermath

While it is tempting to believe that Constantine influenced the decision of the council so that it resulted in a Trinitarian statement, despite only two dissenters (not including Arius) to the Nicene Creed, the emperor's cousin Eusebius of Nicomedia persuaded the emperor to exile the major proponents of the prevailing view of the council. This indicates the Emperor's disfavor with the result of the council and his opposition to the Nicene Creed.

Furthermore, Emperor Valens, one of the emperors after Constantine was a noted Arianist and the persecution of Christianity was resumed under Emperor Julian until Emperor Theodosius I made Nicene Christianity the official state religion in the Edict of Thessalonica. All of these instances provided opportunities for the church to change to Arian beliefs or revive discussions should the Nicene Creed have been published disingenuously or under coercion.

Conclusion

While the actual word ὁμοούσιος (Homooúsios; of the same substance) doesn't have a basis in scripture, adherents to the five solas and Triniteriansm believe that the idea of Jesus being the same as the Father does come from scripture (scriptura) alone (sola).



† Within Ariansim, there were actually three sects, but I have lumped together  ὁμοιούσιος (Homoioúsios) and ὅμοιος (hómoios). These two groups distinguished each other in that the Homoioúsions thought Jesus was of similar substance, while Hómoeans believed Jesus was just similar. They simply didn't buy the philosophy that things had to be created from a substance in the first place, and so their references were void of substance. Functionally however if Jesus is similar to the Father, he is of similar deity of substance ὁμοιούσιος (Homoioúsios) or he is nothing like God and is therefore ἑτεροούσιος (Heterooúsios). As you can see, this clearly breaks down logically, hence why they are lumped together. This helps my second list of 3 correspond to my first list of 3.