[BLU R1 HD Guides, News, & Discussion] :
I found out that you can disable "Amazon Offers", which will result in the lockscreen ads being completely gone. You may notice that going to the Amazon Offers app in the App Info part of the settings results in a greyed out "Disable" button. I was able to fix that. For instructions, skip to the third paragraph.

First, a little more information about the effects of this process. The only ad that remains is a background that reads "Just ask", presumably an ad for the Amazon Echo, or Amazon Alexa. There is no way to interact with the ad, making it effectively a slightly annoying lockscreen wallpaper. This is pretty good, as it removes all the ads and all functionality of the phone is still intact.

In order to do this, I enabled developer options, enabled ADB, connected my phone to a computer, and ran "adb shell". Once on the device, I ran "pm clear com.amazon.phoenix", followed by "pm hide com.amazon.phoenix". This seems to disable the ads, but the app is still enabled, so we're not done just yet. Now I went back into the App Info section of the settings, enabled showing system apps (the option is in the three dot menu in the upper right corner of the apps list), and scrolled down to Amazon Offers. If you completed the last step correctly, the "Disable" button should no longer be greyed out! Go ahead and press it. All Amazon ads should be disabled now, with the exception of the "Just ask" ad that I mentioned before. Fingers crossed hoping Amazon won't be able to re-enable it through an OTA.

While I was at it, I also went ahead and disabled all of the included Amazon apps, with the exception of the Amazon Underground app, because I do use it to shop on Amazon and download apps from time to time. I was also able to disable the Amazon Device Registration app using a similar method, which completely removes the Amazon account integration from the system, and the Amazon Underground app became independent, as it is on any other Android phone. The only Amazon app that I was unable to disable was Amazon Device Information. I don't know what this does, but it seems to be unobtrusive, so I'm not too worried about it. I will keep looking for some indication of what it actually does, and possibly how to disable it.

FWIW: I did not sign into my Amazon account at all prior to doing these steps, so if it doesn't work for you, you might want to try signing out of your account, or resetting the phone if signing out is not an option. Some part of me suspects that signing into an Amazon account before disabling this stuff may somehow complicate things. I did log into my Amazon account in the newly independent Amazon Underground app after disabling all this stuff, and there appears to be no complications.

Update: I'm hearing reports that this does not work on the latest 6.4 OTA. For the record, I have taken that 6.4 OTA, and the ads are not back. That said, I don't remember if I did these steps before or after taking the 6.4 OTA, but it seems likely that I did it before. To summarize, it seems like these steps must be completed before taking the 6.4 OTA, but you can safely apply the 6.4 OTA afterwards and the ads won't come back.

I'm hoping we can find a way to root this thing, and possibly even bootloader unlock it. And hey, why isn't there a forum for this thing? :(

And one final note: I will have limited time to play with this thing, as I actually bought it for my dad. That said, seeing how cheap it is, I'll probably pick up one for myself in a month or two when I have the money, because this is an awesome phone for $60.