What Is This Strange Gmail Notification?
The reasons why you are getting it are unfortunately a tad complicated. You won't be able to to fix it. The problem depends on one's perspective. If you're Google, you think the Optimum doesn't have their server's configured properly. If you're Optimum, you probably have a disagreement over the effectiveness of what Google wants to be the standard for verifying messages sources. I'd have to guess Google will win this battle, and Optimum will implement whatever would make Google happy.
The problem, is that if you do a reverse lookup of the mail servers that Optimum.com uses they reverse to a different domain name (cv.com). Optimum either needs to make the names match up or implement SPF (Sender Policy Framework) to verify the servers identity.
I'll side with Google on this one. Most of the world can manage to figure out how to send email to make Google happy, so Optimum can too. So tell your friend to complain to them!
That said, your friend should really switch to a 'neutral' email address. You don't want to be tied to your Internet provider. Gmail or Yahoo! or Outlook.com would all suffice.
@qrysdonnell: Thank you! That's not difficult to understand, for some odd reason! LOL! My friend won't be changing her e-mail address any time soon -- she's had it forever and she doesn't get these messages, only I do, so it doesn't bother her. Now that I know why I'm getting them, they won't annoy me. I really appreciate your response.
I've had a different Optimum issue, not related to Gmail, but annoying nonetheless. I'm a real estate agent and our company just implemented a transaction management software, called SkySlope for us to use and it's required. The problem is that if I send an email to an optonline.net email, SkySlope says the message is sent, but it never gets received. I've contacted both Optimum and SkySlope,, but to date, no one has a solution. Thank goodness I don't have any current clients with an Optimum email, otherwise I could never send them documents for signature 
Anyone have issues with Optimum mail not being viewable on an Android phone? I don't know if the problem is my phone or Optonline.net.
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Suddenly, in the last month or so, I'm receiving this warning at the top of e-mails I'm receiving from a long-time friend who uses Optimum Online:
"Gmail couldn't verify that this message was sent by optonline.net. Learn more"
I only get this when she initiates an e-mail; when she responds to one of mine, I don't get the notification. I searched all over Gmail and Google and there is no reference to this message (when I click "learn more" it brings me to the issue of spam, but these e-mails are delivered to my inbox, and don't go into spam). She's in my contacts list and, like I said, she's had this e-mail account for as long as e-mails are in existence.
Do any of the tech experts here have any idea why I'm getting this message, and how I can get rid of it?
Many thanks.