Context. Standard stellar evolution theory does not predict existence of Li-rich giant stars. Several mechanisms for Li-enrichment have been proposed to operate at certain locations inside some stars. The actual mechanism operating in real stars is still unknown. Aims: Using the sample of 348 stars from the Penn State - Toruń Centre for Astronomy Planet Search, for which uniformly determined atmospheric parameters are available, with chemical abundances and rotational velocities presented here, we investigate various channels of Li enrichment in giants. We also study Li-overabundant giants in more detail in search for origin of their peculiarities. Methods: Our work is based on the Hobby-Eberly Telescope spectra obtained with the High Resolution Spectrograph, which we use for determination of abundances and rotational velocities. The Li abundance was determined from the 7Li λ670.8 nm line, while we use a more extended set of lines for α-elements abundances. In a series of Kolmogorov-Smirnov tests, we compare Li-overabundant giants with other stars in the sample. We also use available IR photometric and kinematical data in search for evidence of mass-loss. We investigate properties of the most Li-abundant giants in more detail by using multi-epoch precise radial velocities. Results: We present Li and α-elements abundances, as well as rotational velocities for 348 stars. We detected Li in 92 stars, of which 82 are giants. Eleven of them show significant Li abundance A(Li)NLTE> 1.4 and seven of them are Li-overabundant objects, according to common criterion of A(Li) > 1.5 and their location on HR diagram, including TYC 0684-00553-1 and TYC 3105-00152-1, which are two giants with Li abundances close to meteoritic level. For another 271 stars, upper limits of Li abundance are presented. We confirmed three objects with increased stellar rotation. We show that Li-overabundant giants are among the most massive stars from our sample and show larger than average effective temperatures. They are indistinguishable from the complete sample in terms of their distribution of luminosity, metallicity, rotational velocities, and α-elements abundances. Our results do not point out to one specific Li-enrichment mechanism operating in our sample of giants. On the contrary, in some cases, we cannot identify fingerprints of any of known scenarios. We show, however, that the four most Li-rich giants in our sample either have low-mass companions or have radial velocity variations at the level of ~100 m s-1, which strongly suggests that the presence of companions is an important factor in the Li-enrichment processes in giants. Based on observations obtained with the Hobby-Eberly Telescope, which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen. Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Based on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Based on observations obtained with the HERMES spectrograph, which is supported by the Fund for Scientific Research of Flanders (FWO), Belgium, the Research Council of K.U.Leuven, Belgium, the Fonds National Recherches Scientific (FNRS), Belgium, the Royal Observatory of Belgium, the Observatoire de Genéve, Switzerland and the Thüringer Landessternwarte Tautenburg, Germany.Table 5 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/569/A55