What is the scriptural evidence for Satan "taking hold" of Christians' minds or implanting/whispering thoughts?

Identifying the protagonists and antagonists according to Divine Revelation

Always go to the beginning

It is the sixth day of the beginning and God is looking at everything he has made and behold it is very good.

One of the creatures God makes, and in his image and likeness, is man (male and female), and he commands them to be fruitful and multiply, and fill the earth and subdue it.

Even before they get to do that, we are only in Gn 3 and man has fallen out of favor with God at the instigation of the serpent. Man is driven out of garden of Eden and God places the cherubim, and a flaming sword which turned every way, to guard the way to the tree of life.

Divine Revelation - Sacred Scripture and [Holy] Tradition - tells us that these were the only protagonists at the beginning when everything was good: God, his [hosts of] angels, and man. At some point in the beginning one of the hosts and his angels turn antagonists and are thrown out of heaven. The primeval serpent also known as the Devil and Satan is their leader and he is the tempter of Eve [cf. Ws 2:24 (RSVCE)].

There is a war mainly for souls

The Revelation passage above speaks of the salvation and the power and the kingdom of our God and the authority of his Christ.

The protagonist and the antagonists have their realms that are pitted against each other. What is a realm without subjects? St. Paul tells us of how God's Christ has delivered us from the dominion of darkness and transferred us to the kingdom of his beloved Son.

Divine Revelation tells that it is by the sin of our first parents and our personal sins that we fall under the power of the devil - powers of death/gates of Hades - and came to belong to his dominion of darkness. How does the devil rule his subjects? He oppresses them.

The Devil uses every evil and dirty trick that his angelic - now turned devil - mind can conjure up to lure subjects to his dominion and keep them there, be it by tempting to sin, by lying, by the cruelty and persecution of rulers and nations in league with him (cf. Ps 2) , by using the sting of death, by the oppression of demonic possession and obsession [cf. CCC, 1673], etc.

What then is the Devil able to do with those under his power?

Catholic Teaching is that God is always Sovereign, and from the New Advent article on demonic possession:

As we gather from the Fathers and the theologians, the soul itself can never be "possessed" nor deprived of liberty, though its ordinary control over the members of the body may be hindered by the obsessing spirit (cf. St. Aug., "De sp. et an.", 27; St. Thomas, "In II Sent.", d. VIII, Q. i; Ribet, "La mystique divine", Paris, 1883, pp. 190 sqq.).

Answering:

As much as he is allowed by God [cf. the Book of Job], and as much as one has given themselves over to him/his dominion, he will do all he can to keep them under his power and oppression, and in his dominion of darkness. Having thus ensnared them, eventually dragging hapless them down to eternal hell below.

Is there scriptural evidence for or against Satan “taking hold” of our minds or implanting/whispering thoughts?

Yes, and it's worse than you thought!

The other answers have done a fine job at discussing the bible's teaching on direct demonic influence akin to some of the language you use in your question body. I don't discount the reality of this phenomenon at all as I have experienced something along those lines personally. But far more insidious, pervasive and dangerous than what I would call 'overt' demonic attempts (or even successes) at implanting thoughts in people's minds, is the generalized thought implantation process that is conducted at a cultural level - this is a 'covert' operation that unfortunately slips under the radar of every single person born into 'the world'. The Bible says:

...the whole world lies under the control of the evil one. - 1 John 5:19b ISV

And describes how we start out in life as:

In the past you were dead because you sinned and fought against God. 2 You followed the ways of this world and obeyed the devil. He rules the world, and his spirit has power over everyone who doesn’t obey God. - Ephesians 2:1-2 CEV

Most people are familiar with the doctrine of original sin and the Biblical characterisation that:

All of us have sinned and fallen short of God’s glory. - Romans 3:23 CEV

So we will all sin, and (in commonly expressed orthodox doctrine) this is generally blamed on the corrupt 'flesh' that we've inherited from our forebears. What is less commonly recognized and taught is that a large degree of the manifestation of corruption in the flesh stems not from human inheritance, but from demonic influence through the prevailing culture we are born into. In the Ephesians passage cited above, after talking about how the devil rules the world and his spirit has power over everyone who doesn't obey God, the next verse goes on to say:

we were all guilty of falling headlong for the persuasive passions of this world; we all have had our fill of indulging the flesh and mind, obeying impulses to follow perverse thoughts motivated by dark powers. As a result, our natural inclinations led us to be children of wrath, just like the rest of humankind. - Ephesians 2:3 VOICE (original italics)

Can you see how the Bible is saying that as a result of being born into a world system under the control of Satan, our whole thinking process became corrupted so that we were just obeying fleshly impulses and 'perverse thoughts' without regard to God's truth? Many of the common beliefs that pervade our culture (eg. 'It is completely unreasonable to abstain from sex until you're married') are actually Satan's 'best hits' - the lies that directly appeal to our fleshly desires and remain effective tools for enslaving the masses to sin generation after generation. Once a lie like this get's established in a culture, it really doesn't require any direct demonic agency to implant it in our minds - we will quite happily pass the deadly 'meme' around ourselves, cheerfully swallowing it's poison and deceiving ourselves to think that what is evil is good and vice-versa, before regurgitating it at an appropriate moment to infect the next rube that is open for infection.

This is why the Bible places such an emphasis on knowing God's truth and the power it has to transform us:

The Truth Will Set You Free

31 Jesus told the people who had faith in him, “If you keep on obeying what I have said, you truly are my disciples. 32 You will know the truth, and the truth will set you free.”

33 They answered, “We are Abraham’s children! We have never been anyone’s slaves. How can you say we will be set free?”

34 Jesus replied:

I tell you for certain that anyone who sins is a slave of sin! 35 And slaves don’t stay in the family forever, though the Son will always remain in the family. 36 If the Son gives you freedom, you are free! . - John 8:30-36 CEV (Including section title, emphasis added)

No one can have faith without hearing the message about Christ. - Romans 10:17 CEV

Now, if by God's grace we hear this good news of salvation in Christ and come to our senses that God's ways are worth pursuing (and start doing so by grace through faith), we actually have a lot of deprogramming to do! That's why our response to the gospel should be:

Dear friends, God is good. So I beg you to offer your bodies to him as a living sacrifice, pure and pleasing. That’s the most sensible way to serve God. 2 Don’t be like the people of this world, but let God change the way you think. Then you will know how to do everything that is good and pleasing to him. - Romans 12:1-2 CEV (emphasis added)

There are many other verses that are pertinent to this issue, but see in particular:

3 We live in this world, but we don’t act like its people 4 or fight our battles with the weapons of this world. Instead, we use God’s power that can destroy fortresses. We destroy arguments 5 and every bit of pride that keeps anyone from knowing God. We capture people’s thoughts and make them obey Christ. - 2 Corinthians 10:3-5 CEV

and Ephesians 6:10-18 - these 'spiritual warfare' passages give us a very strong sense that the battleground is in the area of our minds - a battle that we need to maintain, and indeed pursue vigorously, even after we have received the saving grace of the Lord Jesus Christ and fully committed our lives to follow him. A battle against demonically empowered ideas that are strongholds in our minds and even in our culture; a fight against a foe who hurls fiery darts of accusation that unless combatted will induce unbelief, fear and condemning thoughts. But we can take courage in this fight because:

...God always makes it possible for Christ to lead us to victory... 2 Corinthians 2:14b CEV

Surrender to God! Resist the devil, and he will run from you. - James 4:7 CEV

In everything we have won more than a victory because of Christ who loves us. - Romans 8:37 CEV

There are several examples in the bible of people who were possessed and/or influenced by devils.

One well known example is in Mark 5:1-20. I'll just quote verse 2.

2 And when he was come out of the ship, immediately there met him out of the tombs a man with an unclean spirit,

And another example here from Acts 16

16 And it came to pass, as we went to prayer, a certain damsel possessed with a spirit of divination met us, which brought her masters much gain by soothsaying:

17 The same followed Paul and us, and cried, saying, These men are the servants of the most high God, which shew unto us the way of salvation.

18 And this did she many days. But Paul, being grieved, turned and said to the spirit, I command thee in the name of Jesus Christ to come out of her. And he came out the same hour.

And another example of possession and perhaps taking hold of thoughts but this time in animals.

Matt 8:28-33

28When he arrived at the other side in the region of the Gadarenes,c two demon-possessed men coming from the tombs met him. They were so violent that no one could pass that way. 29“What do you want with us, Son of God?” they shouted. “Have you come here to torture us before the appointed time?”

30Some distance from them a large herd of pigs was feeding. 31The demons begged Jesus, “If you drive us out, send us into the herd of pigs.”

32He said to them, “Go!” So they came out and went into the pigs, and the whole herd rushed down the steep bank into the lake and died in the water. 33Those tending the pigs ran off, went into the town and reported all this, including what had happened to the demon-possessed men. 34Then the whole town went out to meet Jesus. And when they saw him, they pleaded with him to leave their region.

Was it the demons who drove the pigs into the sea or did the pigs not want the devils in them and so ran into the sea? I don't know. However it is clear that demons can enter into an animal and people.

Calvin has already listed Acts 5:3. Which is a great example too.

I would also point out that the word Satan can also mean adversary. In the case of where Jesus says to Peter in Matt 16:23

Jesus turned and said to Peter, "Get behind me, Satan! You are a stumbling block to me; you do not have in mind the concerns of God, but merely human concerns."

Was Peter possessed by Satan? Well no, because it says in Luke 22:3 "Then Satan entered Judas, called Iscariot, one of the Twelve." and we know that demons are not like God. Satan is not like God and is not able to be omnipresent. Adversary (or one who opposes another) is the meaning here that would fit perfectly with the context.

I love Acts 5 (verse 3):

Then Peter said, “Ananias, how is it that Satan has so filled your heart that you have lied to the Holy Spirit and have kept for yourself some of the money you received for the land? ...

Peter clearly implies agency of Satan here, and apparently, Ananias and Sapphira, while not necessarily believers, were held in that esteem by the early church. I would consider this an incidence of Satan directly prompting actions of believers.

This sounds to me like an instance of implanting new thoughts. I believe this to be the most Biblical supported concept (no quotes as it's a holistic view).