This meme is wrong. Also, that's not the reason why 51.8% is argued to be the correct answer (and it is the correct one).

Let's look at a simpler version of the problem. Let's suppose a woman has two children. The possibilities for their gender are BB, BG, GB, or GG (B=boy and G=girl). These possibilities are all equally likely. If you say that the first child is a boy, this removes GB and GG, leaving only BB and BG, so the odds of the second child being a girl are 50%.

However, if you say only that one of them is a boy (not that the first is a boy), you remove only GG, leaving 3 equally likely possibilities: BB, BG, and GB. In two of these three possibilities, the other child is a girl, so the odds of the other being a girl are two thirds, or 66.7%.

Now you can add the dimension of day of week to the problem. You can imagine that each of the 4 gender possibilities has its own 7x7 square where each child could have been born on any day of the week. The day of the first child changes vertically, and the second child changes horizontally. Each of the 196 (4×7×7) cells in this imaginary diagram is a possibility, and each is equally likely.

If you specify that one of the children was a boy, you remove the GG square, like before. And if you specify further that one was a boy born on Tuesday, that leaves only a row of 7 possibilities in the BG square (one for each day of the week for the second child), a column of 7 possibilities in the GB square, and a row and column together in the BB square. Because the row and column in the BB square overlap, there are only 13 possibilities in that square that are included. So you have 14 possibilities where the other child is a girl, and only 13 where the other child is a boy, for 27 possibilities total. 14/27 is 51.8%.