Yes , I agree with you . It is wrong , but somebody used weight in his research work. Unfortunately, some of his student graduated with Ph.D. using weight instead number .
The researcher may be given weight to the relative abundance of species group (i.e. by computing the sum of the weights of the species, over the number of species). This is a rare and uncommon use.
I must disagree with some of the previous answers offered. There is no reason to use numerical abundances, rather than biomasses, though it is important to only use one or the other (not abundances for some species and biomasses for others) and equally important to never compare a diversity measure based on biomass with one based on abundance. Besides abundance and biomass, it is also possible to use percent cover and some other alternative measures have been proposed. All that is essential is that the data comprise some consistent measure of the relative frequencies of the various species.
This was all set out by E.C.Pielou, in her textbooks published in the 1970s, but can be found repeated in more recent texts. If you do not have access to anything better, you can download a copy of my:
Kenchington, T.J. and Kenchington E.L.R. 2013. Biodiversity metrics for use in the ecosystem approach to oceans management. Can.Tech.Rep.Fish.Aquat.Sci. 3059: vi+188p.
Fish species diversity is the number of different fish species that are represented in a given community (a dataset). The effective number of species refers to the number of equally abundant species needed to obtain the same mean proportional species abundance as that observed in the dataset of interest (where all species may not be equally abundant). Species diversity consists of two components: species richness and species evenness. Species richness is a simple mathematics of species, i.e. count of species, whereas species evenness quantifies how equal the abundances of the species (homegeneity amidst heterogeneity!). But no, there's no space for simple calculation of weight. Biodiversity actually accounts weight of richness, heterogeneity, evenness, no not simple weight of biomass! You need to have the diversity indices concepts in consideration - the species richness, the Shannon index, the Simpson index, and the Gini-Simpson index (=complement of the Simpson index).
The ideas which became biological diversity were first produced by agricultural scientists but much of the early development came from people who worked with insects, using data from the catches of light traps. Since insects have a terminal moult before they can fly, individual weights of adults are essentially fixed and counts of abundance made every sense for the first calculators of diversity. However, once their ideas were taken up by forest ecologists, the latter quickly saw that a count of one sapling is not comparable to a count of one mature forest tree. Hence, they took to using units of weight -- meaning biomass, since a diversity calculation needs a measurement of a species (either count or weight) not a weight of an individual. Marine ecologists working on corals face something of the same problem: counts of colonies, counts of polyps or weights of colonies? Maybe weight of the living tissue on the surface of the colony? Fish ecologists have mostly dodged this dilemma and used abundances but 1,000 larvae and 1,000 commercial-sized adults are not the same thing!
Please do not take my word for any of this. Read Chris Pielou's textbooks and then follow up with the primary papers that she cited. In fact, do _not_ use diversity indices at all if you have not read Pielou. You will almost certainly stray into mistakes if you do not have her guidance.
Better still, do not use diversity measures at all. They looked like a good idea back in the 1960s (though they never achieved what their developers intended) but the arrival of computers powerful enough to handle multivariate statistics of large datasets pretty much put an end to the usefulness of the concept of diversity around 1980. Chris Pielou saw that too. Her last textbook was on ecological applications of multivariate stats.
If you don't like that last piece of advice, then the best index for you to use is probably the Exponential Shannon but, to understand why, you need to read a lot of stuff published in the last ten years. My 2013 Technical Report (see link in an earlier message on this thread) cites much that you should look at but I'm sure there has been more published since.