Yes, connecting the body to the source eliminates the body effect. However, this may or may not be possible in your intended fab process. If you want to connect an NMOS source to its body, and you expect the bodies of different NMOS transistors to be at different voltages, you need to have a process with P-Wells. If instead the process uses a P-type wafer with N-Wells for the PMOS transistors then you can't isolate the bodies of the NMOS transistors.
You may not care, but connecting the sources to the bodies can also significantly increase the capacitance on the source nodes. The well-to-well spacing requirement will also make your layout larger.
The threshold voltage can be increased if the source is not connected to the body terminal. The threshold voltage is
$$V_T = V_{T0} + \gamma\sqrt{2\phi + V_{SB}} - \gamma\sqrt{2\phi}$$
where \$V_{T0}\$ is the threshold voltage when the source-to-body \$V_{SB} = 0\$, and \$\gamma\$ and \$\phi\$ are device parameters. If the NMOS source is connected to ground and so is the body then \$V_{SB} = 0\$ and \$V_T\$ is minimized (it's a similar argument for the PMOS).
So, yes, it is possible to adjust the threshold voltage by not connecting the NMOS body to the negative supply and the PMOS body to the positive supply.
However, this is usually not done intentionally. You typically want to minimize \$V_T\$ -- for example, this would allow you to use lower supply voltages.
The body effect is particularly undesirable for a common drain amplifier because it lowers the voltage gain. Without the body effect the unloaded voltage gain of a common drain amplifier is
$$\frac{v_o}{v_i} = \frac{g_m}{g_m + \frac{1}{r_o||r_{oc}}} \approx 1$$
where the approximation assumes the resistances are large. However, with the body effect the unloaded voltage gain is reduced:
$$\frac{v_o}{v_i} = \frac{g_m}{g_m + g_{mb} + \frac{1}{r_o||r_{oc}}} \approx \frac{g_m}{g_m + g_{mb}} < 1$$
Best Answer
Although in most situations (particularly in instructional settings) this is ignored, FETs are a four-terminal device. The substrate/body of the device acts as a second gate that influences the device behavior. In traditional material this is referred to as the body effect and there are equations to account for how it modifies other device parameters such as the threshold voltage, connecting the substrate to the source is the easiest way to get rid of these equations. In some non-traditional circuit designs this second gate is used to provide additional functionality. E.g., as a lower-gain input to increase the linear range of the circuitry.
In most applications this body terminal must be kept at a potential that guarantees that the drain-body and source-body diodes are kept reverse-biased so that the device can function as a FET. In most cases this can be guaranteed by connecting the body junction to the source, this creates a reverse-polarized drain-source diode. In some power applications, this diode is part of the circuit design and power-FETs include this diode characteristics as part of the available design parameters. In other power applications, such as battery-backup circuitry where reverse polarities are normal, the substrate is actually connected to the effective drain terminal of the FETs.
However, in most IC technologies the substrate itself is the body of the integrated FETs. P-doped wafers are used as a substrate over which NFETs are built. This means that there is only one substrate node for all of the NFETs in the the whole IC. The only way to ensure that the NFET diodes are kept reverse-polarized is to connect the substrate of the IC to the lowest potential in the whole circuit.
For PFETs (which reside in separate N-wells), and in technologies in which both N-wells and P-wells are available, the reason is more about better use of the available space. Connecting the substrate node of such FETs to a different potential, means that the wells have to be kept separated by relatively large distances (the related parasitic devices could destroy the IC otherwise) and that different well contact regions have to be provided. This is done for some designs, but it is better avoided for space efficiency and reliability.