Amplifier gain in budget links
Do you have the noise figure specification of everything downstream of the LNA, and a target system noise figure or noise temperature? If so, you can use Friis' formula to find values of gain and noise figure that are acceptable for your system.
However, in practial circumstances, LNA gain is generally unimportant, again due to Friis' formula. With the equation taken from Wikipedia:
$$F_\text{total} = F_1 + \frac{F_2-1}{G_1} + \frac{F_3-1}{G_1 G_2} + \frac{F_4-1}{G_1 G_2 G_3} + \cdots + \frac{F_n - 1}{G_1 G_2 \cdots G_{n-1}}$$
Where \$F_n\$ is the noise factor (noise figure in linear units) of stage \$n\$, and \$G_n\$ is the linear gain of stage \$n\$.
Note that since the contribution of the noise factor for everything downstream of the first stage (\$n=1\$) is divided by the gain of the first stage and any subsequent stages, the noise factor of everything downstream is basically irrelevant provided that \$G_1\$ is relatively large. Your typical off the shelf MMIC LNA has a gain of around 20 dB, or \$G_1=100\$ in linear units. Unless the noise factor of stages downstream is very, very bad (such as a very long lossy cable run), your LNA is going to swamp out any downstream degredation by a mile.
Given that the typical LNA has a lot of gain, you should be more focused on the noise figure of the LNA. Exactly what maximum noise figure is acceptable depends on the link margin available in your system.
Modern LNAs in the most frequently used microwave bands tend to be very good, with noise figures in the range of 0.5 dB possible if you're willing to spend a bit of money. What often is the limiting factor isn't the LNA noise figure, it's the passive loss upstream of it which degrades the noise figure of the LNA. If everything is at or near room temperature, the noise figure degradation is equal to the passive loss between the antenna and the LNA input. For example, if you have a filter with an insertion loss of 3.5 dB between your antenna and an LNA with a noise figure of 1.5 dB, your actual effective noise figure is 5.0 dB. Usually people over-focus on the LNA itself rather than trying to reduce passive loss. For more details, I suggest reading the Microwaves 101 entry.