Now, there is a weird thing in German grammar (oh yes, another one!) called weak masculine nouns, which get an -n ending when they are declined.
The BBC bitesize page got moved and is now here: https://www.bbc.co.uk/bitesize/guides/z2h4dmn/revision/4 I wish Duolingo would do a little popup window when they slip you a new odd rule of German. I had just got my head round "Die Katze trinkt ihr Wasser" and now this pops up unannounced. Hey ho. At least we get pointers to the excellent forum folk. Thank-you, my fellow Germanophones.
Yes and no. Most of the nouns don't take an -n in the accusative case, not even the masculine ones (it's just the articles or other determiners that take the -n) so this is not the real reason. The reason is that Name, like a few other masculine nouns, belongs to the so-called N-Deklination. These nouns take a final -n in every case (singular or plural) except the singular Nominativ.
By the determiner in front.
kennen takes an object in the accusative case, and seinen must therefore be masculine singular.
Plural accusative would have been seine Namen.
seinen Namen could theoretically also be dative plural, but that doesn’t work together with kennen.
Weak nouns, also called masculine n-nouns, are a group of masculine nouns in German that have a special declension. In addition to inflecting their article, these nouns themselves add an -en or -n ending (-n if the noun already ends in -e) in every case and number except the nominative singular.
Many of the weak nouns refer to people or animals: der Student, der Junge, der Herr, der Nachbar, der Franzose, der Elephant, der Hase, der Affe. Weak nouns that do not refer to people or animals, add an additional -s suffix in the genitive singular.
Ich kenne die Bedeutung seines Namens nicht. "I don't know the meaning of his name."
Or Er wird oft wegen seines Namens gehänselt. "He is often teased because of his name."
Though many people (including me) would be more likely to say Ich kenne die Bedeutung von seinem Namen nicht. with von + dative rather than with genitive, and some would say Er wird oft wegen seinem Namen gehänselt. with dative after wegen, even if that's not grammatically correct in standard German.
Both are correct, with no real difference in meaning.
"Kennen" (and not "wissen") can also be used in the sense of familiarity with the name (e.g., "I've heard your name before, so I know that name"), but obviously that's probably not what is meant here.
Either can be used for just knowing what the person's name is. I believe "kennen" is a bit more common for this usage, but both are fine.
For one thing, the sentence isn't necessarily asking for the first name; the asker could easily be wanting a last name (e.g., "Herr Schmidt").
Even if the first name was clearly meant in context, the German sentence doesn't make a point of explicitly saying that, so there's no reason your English translation should either-- especially since asking for someone's "first name" is kind of an odd way way to ask this in English in the first place.