My friend insists that 'Try the rebooting' is correct in grammar, while the other insists it isn't. Is there anyone who could explain to me what the answer really is? And, please include reliable sources. Thank you!

I would have to disagree with Awesome's analysis. A gerund may be modified by an adjective - including the definite and indefinite articles.

'The closing of the road caused problems.'
'The fruitless searching went on for hours.'
'A quick washing removed the stain.'
'The rebooting of the computer was necessary.'

There are other ways to say all these, but the forms above occur in normal, everyday English.

That still doesn't make the 'Try the rebooting' correct. The problem with that is the wrong article - it should be an indefinite article, not a definite. 'Try a rebooting' is a perfectly understandable, grammatically correct statement.

Note to Beckee: I would have said 'I'm a-fixin' to go to the store here in a bit ...' in my dialect verison of Appalachian - I speak the East Tennessee variety.

Gosh, I don't have many resources to prove this, but...

Rebooting is in the "-ing" form. Therefore it does not require an article (a, an, the)

You could say that in the infinitive:
Try *to reboot.*

You use articles (a, an, the) in conjunction with nouns only.
In other languages, you almost always pair a noun with its article; but English has dropped this habit, leading to confusion sometimes.

So, "a, an, the" ONLY go with nouns.
"Rebooting" is not a noun.
Therefore, "the rebooting" is not proper grammar.

(footing is a noun, but it is defined as such; there are a lot of exceptions in english, but "rebooting" is not one of them)

Rebooting isn't a noun, it's the object (a gerund, actually) of the verb "try." So,

Try rebooting.

The previous answer is a good explanation, grammar-wise. It is a grammatical error, but not a major one. The real problem isn't the grammar, and certainly we understand what your friend means.

It just sounds odd. It's the kind of mistake that, even if I read it instead of hear it, I know it didn't come from a native speaker of English. Native speakers make mistakes in grammar all the time, they just make different kinds of mistakes.

If I were to speak English the way I spoke it growing up, I'd say things like "I'm fixin' to go to the store here in a bit and git some tater chips cuz I ain't gone none." The grammar isn't right, but the grammar isn't the problem.

The problem is that my particular speech pattern identifies me as a hillbilly from Appalachia, and there are certain stereotypes associated with that. People would make assumptions about my family background, socioeconomic status, religious affiliation, educational level, and ability. Some of those would be true, but I find that if people think of me in those terms, they don't always value what I have to say. This really irritates me.

Plus my students have all verified that they wouldn't understand me if I talked like that all the time!