Terminology for a formula like first quartile but using mean instead of median We know:
Def 1. The lower half of a data set is the set of all values that are to the left of the median value when the data has been put into increasing order.
Def 2. The first quartile, denoted by $Q1$, is the median of the lower half of the data set.
Now, suppose the subset of data set which all values that are less than the mean value when the data has been put into increasing order. Then what is this statistical property for mean  value of this subset?

Ref. for Definitions: http://web.mnstate.edu/peil/MDEV102/U4/S36/S363.html
 A: There doesn't have to be a dedicated short name for every slightly different concept. In fact, a little thought shows that would just give us a nightmare, an enormous dictionary-sized list with many special terms. (Arguably, we have that already in that there are several dictionaries or encyclopedias dedicated to statistical definitions, reason enough for trying not to add to the list.) 
Further, much depends on what your readership is already familiar with. There might be little gain in using terms they don't know. So, consider using a short, informative phrase such as "the mean of values below the median" or (to resolve doubts) "the mean of values below or equal to the median" or "the mean of values strictly below the median". Much the same comment would apply to means of values above or below the mean, should they seem interesting or useful. 
That said, a trimmed mean in general is based on setting on one side the lowest $p$% and the highest $q$% of ordered values before taking the mean. Commonly, but not necessarily, $p = q$. Here $p$ is 0 and $q$ is 50, subject to operational detail on whether the median itself is included. 
For a similar, but not quite identical, use of summarising lower and upper halves of the data, see 
Cleveland, W. and Kleiner, B. 1975. A graphical technique for enhancing scatterplots with moving statistics. Technometrics 17(4): 447-454. doi:10.2307/1268431
Examples can be multiplied. Here are two more. 
Means within quantile-based bins are utterly standard in studies of income and wealth. The "significant wave height" in oceanography is the mean of the upper third of the distribution. 
