Why does the input bias current increase when the LMP7721 is powered by a single 5V and the input common-mode voltage shifts by 2.5V?
By definition, the input common-mode resistance is Rin_cm = change_Vcm / change_IB. Thus, for IB not to change at all would require infinite Rin,
However, as you look at Fig 1 above, for 0<Vcm<3V IB changes by only 3fA, which means that the Rin_cm = 3V/3fA = 10^15 or quadrillion ohms (1,000 Tera-ohm). Thus, for all practical purposes Rin_cm is pretty close to infinity and by the same token IB hardly changes.
Having said that, within 2V of positive rail (Vcm>3V on 5V supply) IB increases due to limitations of the bootstrapped circuitry keeping voltage across ESD diodes close to zero - as reverse-bias voltage across ESD diode increases, this results in the increase of leakage current, thus higher IB (see sudden IB roll-off above 3V input common-mode voltage).