The hypercharge in the electroweak model is completely determined by the electric charge of the observed particles. In one usual normalization, the Peskin-Schroder one, it is just the average electric charge of all the particles contained in a weak SU(2) representation. The weak SU(2) representation is defined by the particles that can change into each other by weak interactions, so that the (left-handed) electron and the neutrino are partnered.

Weak SU(2) representations are like spin--- they have one state of each value of "L_z", called I_z, running from -m to m. For the purpose of this discussion, let's assume you have a weak SU(2) spin-2.5 particle, with hypercharge "Y". Then the actual values of the electric charge of the particles will be

-2.5 + Y, -1.5+Y , -.5 + Y , .5 + Y , 1.5 + Y, 2.5 + Y

Y is the offset of the electric charge, and the SU(2) spin value defines the range. The steps are always by one unit. This is the meaning of the formula

$$ Q = I_z + Y $$

For the observed weak partners, the electron-neutrino and the electron, the charges are 0,-1, so the hypercharge is the average value, or -1/2. For the weak partners up-quark,down-quark the charges are 2/3,-1/3, so the hypercharge is the average of the two: 1/6.

But only the left-handed parts of the electron and the quarks are SU(2) partners. The right handed parts have no partner. The right handed parts have a Y which is just their electric charge. The right handed electron has a Y of -1, the right handed up quark Y=2/3, and the right handed down quark Y=-1/3. That's just so that they have the same electric charge as their left-handed partner, so that they can form a massive charged particle together.

One natural normalization of hypercharge is not Wikipedia's nor Peskin Schroeder's. This is in terms of the greatest rational value for which gives all the standard model particles integer hypercharges. This value is 1/6. In terms of multiples of 1/6, all the standard model particles have a hypercharge of 1,2,3,4 and 6 units.

But this assumes that the U(1) of hypercharge, with its crazy values, is fundamental, which is extremely unlikely. The most natural normalization choice comes from embedding SU(2) and SU(3) into SU(5) (or a higher GUT of the same type, like SO(10) or E6). In this embedding, you think of SU(5) as a 5 by 5 matrix, the top 2 by 2 block is the SU(2), the lower 3 by 3 block is SU(3), and the U(1) consists of all diagonal phase-matrices (a,a,b,b,b) where a^2b^3=1, so that this phase is generated by

$\mathrm{diag}(1/2, 1/2, -1/3, -1/3, -1/3) $

Meaning that if you rotate by the diagonal matrix with $(e^{i\theta/2},e^{i\theta/2}, e^{-i\theta/3}, e^{-i\theta/3}, e^{-i\theta/3})$ on the diagonal, you are still in SU(5), but you are not in SU(2)xSU(3).

The matter is the defining representation of SU(5), plus the antisymmetric two-tensor representation. The decomposition of these two representations explains the hypercharge assignments of the standard model easily and naturally. See this answer: Is there a concise-but-thorough statement of the Standard Model?