Is there a way to send a (sms) text to myself when I receive a new Gmail message and the subject contains a certain string?
Is there a feature in Gmail itself, a third party service, or an addon that would help me do this?
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IFTTT (If This Then That) may be able to help you out. You can find Connect Gmail to SMS recipies
EDIT: As noted in the comments section, IFTTT stripped much of the Gmail functionality from their service in 2019, meaning this solution no longer works.
One other potential solution would be to use Zapier as they have some Gmail and SMS integrations, but it says that "only business accounts" will work for said feature.
Since cloud services may change, it might be more opportune to create your own service to accomplish this task:
If you are comfortable with writing code you could look into using Google Apps Script and AJAX requests with a PaaS which provides SMS communication APIs, such as Twilio, to accomplish a similar task. Unfortunately, from what I found there is not a simple installable trigger for when an email comes in; therefore you will need to write logic around a time trigger that will then process said email. This solution seems to work well for others (though I haven't tested it); as for hooking up Twilio to Apps Script, there's an official Google Video for how to do so with Google Sheets.
I don't know of a way to get it to send an SMS, but there is a solution if you have an Android phone.
I have my Gmail set with a filter (in settings-filters) to set a label ("notifyinandroid") for certain senders, then I have the Gmail app on my phone set to only notify me when items arrive with that label. You could also do this for certain subjects.
That way my phone will notify me if I have an email from somebody that I care about. This setup works great for me.