A few colleagues told me how they taught their kids to sign and how it reduced frustration, so I looked it up, my notes are here. A lot of the papers claiming benefits from sign language seem to be from the same few people, and when I looked for the opinions of others, I got:

Claims that signing with infants benefits language development are examined. Fourteen infants aged 19 to 23 months were tested on their comprehension and production of novel labels in a word learning task. Infants participated in two conditions. In the Sign + Word condition, infants learned both a signed and vocal label for a novel toy, whereas in the Word Only condition, infants were taught only a vocal label for the novel toy. Results showed that when children participated first in the Sign + Word condition, their comprehension and production abilities were lower than when trained first in the Word Only condition. Previous exposure to sign language was not related to infants’ performance on the word learning task, although there was a marginal effect of previous language ability on performance. Contrary to previous findings (e.g., Goodwyn, Acredolo, & Brown, 2000), the sign and word combination did not facilitate children’s learning of spoken labels.

... so it may have some small benefit, but nothing huge.

We tried teaching ours a few simple signs, the only ones that stuck were those for eating and drinking. Now he's speaking a bit, so there's not much point any more.