How do you name #"H"_3"C"("CH"_2)"OCH"_3#?

In this case there is no need to look for the longest chain; it's an ether, no? Just draw it out from left to right, because you were given the structural formula.

#color(green)("H"_3"C"-stackrel("H")("C"))-stackrel(..)("O")-color(blue)("CH"_3)#
#color(white)(aaaaaa)^color(green)("H") color(white)(. ..)^(color(blue)(..))#

(Note that the molecular geometry around the oxygen should be bent.)

For an ether, you can choose the alkyl groups on either side of the oxygen and name them individually.

So, this would be ethyl methyl ether because there is an ethyl on one end and a methyl on the other. (Apparently you can also call it methyl ethyl ether, but I don't really like that because it isn't alphabetical on the substituents that way.)

Another way you can name it is if you consider the #"O"-"CH"_3# to be a functional group attached onto a two-carbon main chain. Then, the #"OCH"_3# is called a methoxy group.

#"H"_3"C"-stackrel("H")("C")-color(blue)(stackrel(..)("O")-"CH"_3)#
#color(white)(aaaaaa)^"H" color(white)(. ..)^(color(blue)(..))#

Thus, an alternate name is methoxyethane (but not ethoxymethane, because ethyl is longer than methyl).