A thin blue line on my iPhone 4S?

Did you also try to reset the phone by holding the sleep and home button for about 10sec, until the Apple logo comes back again?

You will not lose any data by resetting, but it can cure some glitches.

If this does not help,and to rule out a software problem, setting it up as new device would be the next step:

Use iTunes to restore your iOS device to factory settings - Apple Support

If still no luck, the phone needs to be serviced, but since Apple does not offer to replace the screen for an iPhone 4s, you would have to pay the service fee to replace the whole device.

Out-of-warranty repair service
If your repair isn’t covered by Apple’s One Year Limited Warranty, AppleCare+, or AppleCare Protection Plan, your iPhone might be eligible for out-of-warranty service. For example, liquid damage isn’t covered by warranty but might be eligible for out-of-warranty service. Some damage isn’t eligible at all, for example if your device has been broken into multiple pieces. See Apple’s Repair Terms and Conditions for complete details.
Read Apple’s Repair Terms and Conditions

Model	Out-of-warranty service fee	Battery service*
iPhone 6 Plus	$329	$79 *available only if battery fails Apple’s diagnostic test
iPhone 6	$299
iPhone 5s, iPhone 5c, iPhone 5	$269
iPhone 4s	$199
iPhone 4, iPhone 3GS, iPhone 3G, Original iPhone	$149

Plus a $6.95 shipping fee, if required. Fees are in USD and exclude local tax. Pricing is for service through Apple. The final service fee we charge will be determined during testing and may be less than the service fee listed above. Pricing and terms vary for service through an Apple Authorized Service Provider.

copied from Service Answer Center - iPhone

It looks like a set of bad pixels on the screen. That would also explain why it does not appear in the screenshot. The screenshot is actually from the video feed to the screen, not the actual screen itself. I agree with ingo2711 and recommend an appointment with the Genius Bar.

Apple does not offer trade ins for older devices, they offer a recycling program, but you'll get more for your old phone, if you try to sell the phone yourself.

Since opening the phone will void the option to get it replaced by paying the Out-Of-Warranty fee, any attempt to get the screen replaced by a 3rd party repair center will be on your own risk, or the one who is willing to buy a phone with a defective screen.