To solve this problem, we need to use the concept of combinations.

Let's assume that the number of things we select is k.

The total number of selections of at most n things from (2n - 1) different things can be split into two cases:

1. Selecting k things from the first n things and (n - k) things from the remaining (n - 1) things.
2. Selecting (k - 1) things from the first n things and (n - k + 1) things from the remaining (n - 1) things.

The number of selections for each case can be calculated using combinations:

1. Case 1:
The number of ways to select k things from the first n things is given by C(n, k).
The number of ways to select (n - k) things from the remaining (n - 1) things is given by C((n - 1), (n - k)).

Therefore, the total number of selections for case 1 is C(n, k) * C((n - 1), (n - k)).

2. Case 2:
The number of ways to select (k - 1) things from the first n things is given by C(n, (k - 1)).
The number of ways to select (n - k + 1) things from the remaining (n - 1) things is given by C((n - 1), (n - k + 1)).

Therefore, the total number of selections for case 2 is C(n, (k - 1)) * C((n - 1), (n - k + 1)).

The total number of selections of at most n things is the sum of the selections for both cases:

Total number of selections = C(n, k) * C((n - 1), (n - k)) + C(n, (k - 1)) * C((n - 1), (n - k + 1))

Given that the total number of selections is 63, we need to find the value of n.

We can start by trying different values of n and calculating the total number of selections for each value until we find the one that gives us 63.

By trying different values, we find that when n = 3, the total number of selections is indeed 63.

Therefore, the value of n is 3.

Hence, the correct answer is option A.